Perhaps you remember learning the FOIL Method back when you were in Algebra I.
To refresh your memory (in case you blocked it out), this crappy method taught you how to multiply binomials.
To begin, you multiply the Firsts, which would be the first term of each binomial. Then you multiply the Outers. Then you multiply the Inners. Then you multiply the Lasts. The result is the product of the two binomials. Boring!!
In my opinion, the process should not be taught, learned, studied, or considered. Why bother with an algorithm that applies only to one operation (multiplication) and only to one type of factor (binomials)?
I would like to suggest to math teachers everywhere that the average person (who can add, subtract, multiply, or divide regular constants) is well-equipped already with the schemata to successfully operate with polynomials of ANY size.
A few years ago, I mentioned this strategy at a Math Department meeting, and my colleagues loved it. They had never considered it!
Rather than belabor a description of it, here's some how-to visuals which I hope will make sense. Email me at ThatCatholicGirl@Catholic.org if you have any questions!