I have taught math for many years (too many?) and one thing that I have found over the years is that kids have trouble multiplying higher order polynomials. Teachers all have their favorite methods of teaching the process. I have made it my goal to learn as many of the methods that I could. (Some just have made no sense to me so I cannot say that I have actually learned them all.)
The problem, as I see it, is that we (math teachers) keep teaching binomial (only 2 terms) multiplication for days on end first. Then as an after thought throw in a few higher order problems and expect the kids to extrapolate knowledge that they don't really have.
The kids memorize a procedure--the favorite method around here is F.O.I.L.--and they practice it. FOIL only works for binomials. It does not have enough letters in the name to allow it to be used as a mnemonic for trinomials or beyond.
I would not even acknowledge the mnemonic except that the kids have all heard of it before they get to me. I prefer to work on the higher order problems first. Then the kids can extrapolate their skills back to the simpler binomial problems.
Step 1: F.O.I.L.
Before we get to my 2 favorite multiplication methods, let me take a moment to explain FOIL for anyone who might not have seen it.
'F' stands for first. You multiply the first thing in each of the sets of parentheses.
'O' stands for outside. You multiply the outer most extremes.
'I' means inside. You multiply the inner most terms.
'L' refers to the last terms. You multiply the last term in each of the sets of parentheses.
The last thing you do is combine any like terms--usually the terms from the 'outside' and the 'inside'.
If students saw this as an extension of the distributive property (which they also apply without fully understanding), they would be able to extend the process to allow for more terms in the polynomials. Sadly they don't and they can't. So we are left with 2 options: give up (which is not an option) and finding another way to explain the process.
Step 2: Alternative #1: Traditional Multiplication
When most of us were in school, we were taught the same method of multiplying 2 numbers. My son's were taught the 'lattice' method so I know some people learned different methods first. The middle school teachers tried to un-teach it--to varying degrees of success. In college, I actually learned 5 or 6 other method of multiplying. I still go back to the traditional method that I learned first.
This method works well for polynomial multiplication. Write the longer (more terms) polynomial on top with the shorter one under it. Line up the terms like you would if they were just individual numbers. Pic #1
Multiply the bottom right term by each individual term of the polynomial above it. Start on the right hand side only because that is where order we learned as kids. Make sure to leave a space between terms (or have the kids insert a + or - sign). I have found that kids will insert appropriate minus signs but usually forget the plus signs. A space is easier to remember. Pic 2
When that row is complete, move to the next term to the left on the bottom polynomial. Multiply it by the right hand term of the top. Before you have students write it down, have them look at the first row of their answer. Find the 'like' term. This marks where they should place the new term they just found. When they multiply the second term on the bottom by the second term of the top, they will probably find that they need to place it just to the left of the previous term. This feels 'normal' to those of us that grew up on this method of multiplication. Pic 3 and 4
Repeat this process for each term of the bottom polynomial if there are more terms. Make sure that 'like' terms are always lined up properly.
Then you draw your equal sign (line under all your work so far) any you add the columns. Now they must put in the appropriate plus signs to separate the terms. Pic 5
This was my go to method for a lot of years. Most kids could work this way. Most--not all.
Step 3: Alternative #2: Lattice Multiplication
A few years ago, when the 'every day math' kids made it to the high school, I added this method to my algebra classes. Every Day Math was the name of the program that the elementary schools adopted that uses the lattice method to teach multiplication. This method made more sense to a lot of kids. I told them that they could use whichever method they preferred, once they had tried both methods.
Start with the grid--it doesn't matter which polynomial you put across the top but I usually put the longer one there (saves a bit of paper). I usually put the second polynomial on the right but I think that is because I am right handed. You do not use the diagonal lines that get drawn into the grid for regular lattice multiplication. Pic 1
Pick a row to start with. Multiply one term by one term. Put the 'answer' in the row/column box that corresponds to the row and column that you got the terms from. You are done with this step when your entire grid is filled in. Pic 2, 3, and 4
Now, all you have to do is combine the 'like' terms. They will usually be located on diagonal lines from each other. Pic 5
Once the students can do these problems, something like (3x + 5)(7x - 6) is almost insulting because it is so simple.