
This page explains FOIL, which means First, Outer, Inner, Last.
It's the
method for multiplying two expressions, each of which has two terms.
How to recognize a F.O.I.L. situation 


First, I'll remind you of the distributive property of addition over multiplication. It states:
The sum of A and B, multiplied by any number or expression E is equal to the sum of A multiplied by E and B multiplied by E. In symbols, it can be stated as follows:
(A + B) (E) = AE + BE
EXAMPLE

Now, if E = C + D, we can substitute C + D in place of E in this equation, as follows:
(A + B) (C + D) = A (C + D) + B (C + D)
Now, the distributive property can be used two more times as follows:
(A + B) (C + D) = AC + AD + BC + BD
Notice that AC (the product of A and C) are the first terms, AD are outer terms, BC are inner terms, and BD are outer terms  first, outer, inner, and last, or F.O.I.L.
Now, let's return to our first example:
EXAMPLE

The first terms are 3x and 2x. Their product is 6x².
The outer terms are 3x and 3. Their product is 9x.
The inner terms are 8 and 2x. Their product is 16x.
The last terms are 8 and 3. Their product is 24.
Adding them up, the sum is 6x²  9x + 16x  24.
Simplifying, the sum is 6x² + 7x  24.
If you're smart, you have already figured out that simplifying using F.O.I.L. is the opposite of factoring. In other words, if you are asked to factor the trinomial 6x² + 7x  24, you will reply (3x + 8) (2x  3). If you are asked to simplify (3x + 8) (2x  3) using F.O.I.L., you will reply 6x² + 7x  24.
The webmaster and author of this Math Help site is Graeme McRae.