The FOIL method is an acronym for First, Outer, Inside and Last. This just means that it is the order in which a binomial will be multiplied and then solved.
A binomial is just the name for two terms. For example: (5x-3) is one term and (2x+7) is one term.
When we see them next to each other (5x-3)(2x+7), we have to multiply them using the distributive property to find the answer. This is where the FOIL method comes in to play.
Remember when you multiply any numbers together, you MUST use the sign in front of each number when you do!
A positive number times a positive number = a positive number answer
A positive number times a negative number = a negative number answer
A negative number times a negative number = a positive number answer
Let's Use the FOIL Method on the above binomial: (5x-3)(2x+7)
1) First means multiply the first two terms in each parenthesis, which will be the 5x and the 2x; they are both positive, so the answer is a positive 10x
2) Outer means multiply the two terms that are on the outer most side of each parenthesis, which will be the 5x and the 7; they are both positive, so the answer is a positive 35x
3) Inside means multiply the two terms that are on the inside right next to each other, which will be the -3 and the 2x; the first is negative and the second is positive, so the answer is a negative - 6x
4) Last means multiply the last two numbers in each set of parenthesis, reading each set separately from left to right, which will be the -3 and the 7; the first is negative and the second is positive, so the answer is a negative - 21
Now line up your answers and then combine the like terms to find the complete answer. Don't forget to take the sign with you when you line them up and to work left to right!!
10x + 35x - 6x - 21
1) 10x + 35x = 45x
2) 45x - 6x = 39x
3) There isn't anything to combine with -21, so it just remains -21
The final answer then is 39x - 21
I hope that this helps!