Multiplying Polynomials
A polynomial looks like this:

example of a polynomial
this one has 3 terms 
To multiply a polynomial:
 multiply each term in one polynomial by each term in the other polynomial
 add those answers together, and simplify if needed
Let us look at the simplest cases first
1 term × 1 term (monomial times monomial)
To multiply one term by another term, first multiply the constants, then multiply each variable together and combine the result, like this (press play):
(Note: I used "·" to mean multiply. In Algebra we don't like to use "×" because it looks too much like the letter "x")
For more about multiplying terms, read Multiply and Divide Variables with Exponents
1 term × 2 terms (monomial times binomial)
Multiply the single term by each of the two terms, like this:
2 term × 1 terms (binomial times monomial)
Multiply each of the two terms by the single term, like this:
(I did that one a bit faster by multiplying in my head before writing it down)
2 terms × 2 terms (binomial times binomial)
Each of the two terms in the first binomial 
is multiplied by 
each of the two terms in the second binomial 
That is 4 differrent multiplications ... Why?

Matching up Partners
Two friends (Alice and Betty) challenge two other friends (Charles and David) to individual tennis matches. 
How many matches does that make?
 Alice needs to play Charles, and then Alice needs to play David.
 Also Betty has to play Charles and then Betty has to play David.
They could play in any order, so long as each of the first two friends gets to play each of the second two friends. 
It is the same when we are multiplying binomials!
instead of Alice and Betty, let's just use a and b, and Charles and David can be c and d:
You can multiply them in any order so long as each of the first two terms gets multiplied by each of the second two terms.
But there is a handy way to help you remember to multiply each term called "FOIL" that stands for "Firsts, Outers, Inners, Lasts":

 Firsts: ac
 Outers: ad
 Inners: bc
 Lasts: bd

So you multiply the "Firsts" (the first terms of both polynomials), then the "Outers", etc.
Let us try this on a more complicated example:
2 terms × 3 terms (binomial times trinomial)
"FOIL" won't work here, because there are more terms now. But just remember:
Multiply each term in the first polynomial by each term in the second polynomial
