If I'm tossing 4 pennies at once, and then recording how many heads there came out to be 32 times, is that a Binomial experiment?
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No, because each trial (tossing 4 pennies at once) can result in more than two possible outcomes. This is according to a definition of "binomial experiment" on google, but if the term is used loosely enough then I am sure that anything involving flipping coins can be called a binomial experiment.
Edit: I guess that answer was too pedantic and that the real answer to your homework question is probably "yes". If I understand your question correctly, you are counting the total number of heads out of 128 coin flips, where you are flipping four coins at a time. Unless you have a very strange model (for example the coins interact somehow in flight), this is the same experiment as flipping the coins one at a time, in the sense that the total count of heads has the same binomial probability distribution. So maybe a better answer is "technically your experiment is not a binomial experiment because each trial has more than two possible outcomes, but the distribution of the total count of heads is the same as that of the binomial experiment where you flip the coins one at a time, so in that sense your experiment is a binomial experiment."
The number of heads you get each time you toss four pennies is either $0$, $1$, $2$, $3$, or $4$, and its probability distribution is a binomial distribution. If I understand you correctly, you're doing that 32 times, and recording each time how many heads you get. However, the term "binomial experiment" usually means something with only two possible outcomes. So when you toss one penny, that's a binomial experiment.