Aside from very simple or colonial organisms, all animals on Earth have only one mouth for eating. Many organisms have multiple spiracles or even multiple anuses, but not mouths.
The problem isn't the anatomy or mechanics - life will find a way to make it work. The problem is figuring out a reason why such a system would evolve in the first place. Despite the many, many varied body plans in life on earth, there is not a single creature (not counting colonial organisms or unstructured animals like sponges) with more than one mouth, and every animal that has a head has a mouth on it. There simply aren't enough benefits to outweigh the costs of such a body plan.
On earth, the formation of the digestive system is one of the first parts of nearly every animal's development - a ball of cells develops either a cavity (for animals with one orifice, like jellies) or a tube (for those with both a front and a back orifice) and the creature develops from there. The basic structural differences between any vertebrate and a foot-feeding centaur are significantly greater than the differences between a human and a jellyfish.
That being said, let us consider a way it might happen.
On a planet where such creatures evolved, you'd need to start out with a completely different body plan - one that developed multiple feeding tubes instead of one. A primitive animal on such a world might loosely resemble a starfish, but one with mouths on its legs instead of in the center. The only way such a system would make sense is if the earliest animals were sedentary, and used multiple tentacles to grab at food particles, possibly sucking them up as they floated past.
Over time, the creatures could have learned to use their tentacles for motion, developing them into fins or legs while retaining the mouths at the ends. However, it would still seem likely that, were the creatures to become fast-moving organisms that pursued prey or floating food particles, it would make much more sense for one of the tentacles to develop into a specialized head with sense organs and a mouth while the mouths in the other tentacles atrophied. Therefore, the only way for this system to be retained plausibly is if the creatures were always slow-moving grazers, perhaps beginning their history feeding off of bacterial mats on the sea floor, while using one specialized tentacle to watch out for predators.
Grass would have had to show up on their world very early to have kept the exclusively grazing lifestyle viable after the creature's ancestors left the seas (on earth grass only showed up after the dinosaur age - early earth herbivores were browsers, not grazers) and keep the creatures from adopting the much simpler body plan of a single head with a mouth and eyes that looked for food and ate it. However, I would expect that, since the purpose of the head is to look out for predators, I would expect it to be in the center and have eyes positioned all around it, not at the front. What could this creature be pursuing that would cause it to benefit from front-facing eyes and a head at the front of its body?
The hands could have been a later development, probably for fighting off predators or intraspecific competition, since climbing would not benefit these ground-feeders much. It could be that it developed a distinctive front and back end in the first place so that it could fight more effectively, but never actually ate the things it was fighting with.
Altogether, this is a very, very unlikely body plan, even for an alien.
What you are talking about has been talked about in the A Natural History page featuring Centaurs and instead of blatantly repeating the words he wrote, I have instead included two scans of the book and a transcript from the original online post.
As the author of the book states "Since centaurs cannot easily stoop their upper body to reach low lying plants they rely on a peculiar method of ingestion to eat while on the move. Centaurs graze through grasping mouths on the base of their four hooves. These toothed orifices are attached to a length of esophagus-like tissue that runs the length of the centaur's limbs. The process appears to be automatic, and centaurs continue to graze even while they sleep.". A close up of the foot shows exactly what he mean, the throat running up the leg
When I think of a creature with multiple eating mouths, I tend to think of a creature like the Lernaean Hydra, which has multiple heads sprouting from a common body. Such a creature, in my opinion, is likely to evolve in an evironment where there are a lot of creatures that "steal" kills from predators, making quick, efficient eating of your kill highly advantageous. Here's my idea on how such a creature could evolve.
Imagine our proto-hydra: a small, reptilian creature, with the unusual adaptation that the brain is located within the body, instead of the head. This makes the proto-hydras head little more than a highly-mobile sense organ with a mouth attached to it. It obtains food through scavenging the kills of larger predators, usually darting in while it is distracted and stealing a few bites before scurrying of.
Then, a proto-hydra is born with a birth defect: due to a genetic mutation, this proto-hydra has a second head and neck! Since the brain is within the main body cavity instead of the head, this second head does not develop its own desires or plans, like with two-headed versions of other species*. Thus, this proto-hydra has an advantage: it can eat twice as much food in the same amount of time! In its well-fed state, it is better able to compete for mates, and while some potential mates reject it due to its difference, enough are impressed with its size and strength, and allow it to mate with them. Thus, the gene for multiple heads is passed on, causing a new species of proto-hydra to diverge from the first, this second species having two heads!
This process can happen repeatedly throughout the eons, causing the proto-hydras to develop more heads. At the same time, the larger size given to them by their scavenging success eventually allows them to graduate from scavengers to predators. First with small prey, them, when they continue to grow larger, bigger prey. Then, the multiple heads allow the proto-hydra to quickly devour its kill, leaving nothing for the scavengers, along with possibly leaving a head up to watch for and predators that might try to eat it.
After some more evolution, adding other hydra traits like spitting venom and the ability to regenerate lost heads, you would have the hydra, a creature with multiple eating mouths, because of the need to leave no leftovers for anyone else.
*In cases of a carnivorous species developing a second head, like snakes, there have been known cases of one head trying to eat the other!
While I agree with the others that the body plan you suggest is unlikely, a slight alteration would make it work.
Have mouths in the hooves as suggested, however nothing behind them. They just grab the food and chew it, storing it in a small area around the size of a mouth and with a few enzymes similar to our saliva.
The actual "mouth" would be in the center of the body. The hooves grab the food and then deliver it to the "mouth" after chewing it just by bending up and pushing it in.
This could be explained as evolution from a starfish like design, with grasping limbs and a central mouth. One limb evolved into the sensing "head" and the others for grasping and locomotion.
I'll take a slightly different approach to this than the others, I think.
Consider such a creature with two eating mouths at two different heights. Perhaps this creature is very tall, with one mouth to eat fruit from tall trees and another to eat fruit from lower bushes or meat from prey. To make such a system required, one mouth can't perform both tasks. I'm thinking we make this necessary by giving the creature a severe inflexibility so the high mouth can't be reached by the arms / tentacles / grabbing appendages down low to be fed, which also necessitates another set up near the high mouth (unless we give it a shortish, prehensile neck and it just grabs the fruit with its mouth).
Now we run into the problem of why such a body structure is necessary. This is where we look at the world in which the creature lives.
First, it needs a high food source and a low food source. Think giraffe which can't reach the ground with its head. Why does it need to go low if it can already go high like a giraffe? Perhaps the fruit in this world doesn't provide enough nutrients to sustain life on its own and must be supplemented by something that can't be found up high.
But why go high in the first place, then? Perhaps we say the creatures on this world require both forms of sustenance to survive--one which can be found in fruit, one which can only be found in ground-level plants or other animals.
Perhaps there was some cataclysm which mutated all life on the planet, and only the mutations which could adapt to get both kinds of sustenance could live (vaguely reminiscent of Orson Scott Card's Speaker for the Dead).
There are still holes in this scenario, but I think you'd need to look at the environment in which the creature lives to require it to have two feeding mouths.
- A biome where the animal's food source becomes available for a VERY brief period and is somewhat spread.
- Start with a giraffe.
Now imagine that the only trees in the area have flowers/leaves that are inedible at all times besides an extremely narrow window: perhaps less than a minute. In order to consume enough to survive, the giraffe may need to develop multiple heads so that it can eat sufficient quantities before its food goes back into hiding. Perhaps a world where light rarely and briefly shines through to these plants, which open just long enough to drink it in. When not eating, the creature uses its mostly damaged/inferior heads for self-defense using an aggressive biting style.
Here is a real-life example showing how multiple mouths are possible even with the bilaterally symmetrical body plan.
Evolution is the ultimate cost/benefit shopper. It costs energy to make a redundant body part, so that body part better be worth keeping.
A body plan with multiple mouths would evolve when you need to cover more ground (e.g., foraging) or eat very quickly (e.g., feeding frenzy) or you are likely to lose one mouth and need backups (e.g. lungs will pop) or the cost of making of an extra body part possibly confers some advantage not directly related to its primary function (e.g., huge breasts don't feed babies better than regular breasts but attract more mates).
One interesting idea is "stereo." You need [at least] two eyes for depth perception, and two ears for stereo to tell where a sound is coming from, and two nostrils to tell who farted (though this last one works better for dogs). The heads of hammerhead sharks are shaped that way because in order to increase distance between their electroception organs, and thus get a better targeting read on their prey.
Maybe a pair (or more) of mouths can actually double as smell receptors too (think snake's tongue), so as to more accurately locate food?
Let us consider an organism that consumes two categories of food: extremely acidic, and extremely alkaline foods. No one set of digestive organs can processes both foods, and we don't want them to react with each other inside the creature's body. So it evolves two digestive tracts, which are each optimized for either alkaline or acidic foods.
If a trait is successful it allows a creature to produce more successful offspring.
That being said some traits are not only good for single solutions but multiple solutions. That two mouth body plan becomes more believe the more solutions it generates.
Teeth offer, in one form another, are good for eating (we'll call it rigid food processing), are also good at biting competitors, warding off predators / rivals, in some cases digging holes, in some cases articulating speech (quacking, barking, etc). Therefore teeth should be found in great abundance: https://www.affiliateddentists.com/info/animal-teeth-sea-creatures/
Likewise bilateral symmetry is incredibly useful. One limb (a snake or earthworm could be considered single limbed for the purposes of this discussion) is great for some applications (slithering, hiding, etc.). However damage to one limb is damage to all. Bilateral symmetry provides a great backup system. Additionally two limbs can work together with possibly greater efficiency than one (swimming, locomotion).
More reasons for two mouths are better than one as long as environmental pressure supports multiple reasons. Coping with a highly predatory environment (food stealing), digesting two types of food (two different enzymes?), etc.
Finally, consider the rest of the environment. Some strategies are successful enough to express across kingdoms. Such as eusocial social structures which occur in insects (ants, bees, termites), mammals, (naked mole rats), and shrimp (https://en.wikipedia.org/wiki/Eusociality).
I agree with IndigoFenix. Before I saw Xplodotron's answer, I could think of only one creature that has multiple mouths: Venus Flytrap.
Granted, that uses the term "mouth" pretty loosely but they do trap and digest prey. They also die when they take in prey and are replaced by new traps.
Also, as Xplodotron mentioned, sexual selection may allow a random mutation that does not decrease survivability too badly. A two headed snake as a pretty common mutation. If the multiple heads helped the male snake mate it might get passed on.
How about a different approach:
Lets go down the evolutionary tree to a very primitive basically immobile creature with the usual tube based digestive system. This creature has an asexual reproduction mechanism in addition to the usual sexual one--hence it can have offspring that are genetically identical to itself.
Now, one of these creatures learned how to cooperate with offspring that had not wandered off. Eventually this cooperation became so complete that they fused into a single organism. (Remember, genetically identical, this wouldn't be unreasonable.) Now you have a single creature with multiple digestive tracts that can serve as the base for more complex multi-mouthed creatures to evolve. There must be some considerable advantage to it, though, or it's going to lose out to creatures without the duplicated digestive apparatus.