Understanding what you have when you have a polynomial is one thing, but performing operations like addition, subtraction, multiplication, or even division is another. Today, we will just talk about a special case that comes up when you multiply two specific types of polynomials – two binomials.
Remember, a binomial is a polynomial with two terms that are either added or subtracted from eachother like 
First: multiply the first terms of each binomial
Outer: multiply the “outer” terms in each binomial
Inner: multiply the “inner” terms in each binomial
Last: multiply the “last” terms in each binomial
If you think about it, this does in fact make sure every term is multiplied by every other term. Take the first term for instance. Once you have FOILed, it has been multiplied by the first term of the second binomial and the last (by the “outer” step). This is true for every term!
Once this process is complete, it is very likely you have some cleaning up to do in the sense of combining like terms, so you should make sure to simplify your answer once you are done. Here is how it works with an example:
First: The first two terms are the 
Outer: The “outer terms” are the 
Inner: The inner terms are the 
Last: The last terms are the 
The final answer would be the terms we got in each step above added together. The steps all together would look like this:
This is also the same method you would use to find the square of any binomial like 
A last but important point: This method will NOT work if one of the parentheses contains more than two terms (in other words if either is not a binomial). For instance, to find 
Also, if you have more than one binomial, FOIL may not work by itself – you may need to use FOIL on the first two binomials and then some other method depending on the results of that. This is a very nice method but make sure you apply it only when it makes sense!
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One way to avoid the need for defining this only for 2 binomials, is to teach it as an application of the distributive property instead.
If you think of multiplying two binomials as just applying the distributive property twice, it will extend to trimomials and other polynomials much easier.
A trinomial times a trinomial is simply applying the distributive property THREE times. No need for gimmicks.
Very nice way of approaching this! It is interesting that “FOIL” has become so ubiquitous when it is a special case of a (likely as you pointed out) easier to understand general idea. I’m not sure of the history on that at all though.