Multiplying Polynomials
A polynomial looks like this:
example of a polynomial this one has 3 terms 
To multiply two polynomials:
 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 PartnersTwo friends (Alice and Betty) challenge two other friends 

How many matches does that make?
They could play in any order, so long as each of the first 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:
We 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 us remember to multiply each term called "FOIL".
It stands for "Firsts, Outers, Inners, Lasts":

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
Like Terms
And always remember to add Like Terms:
Example: (x + 2y)(3x − 4y + 5)
(x + 2y)(3x − 4y + 5)
= 3x^{2} − 4xy + 5x + 6xy − 8y^{2} + 10y
= 3x^{2} + 2xy + 5x − 8y^{2} + 10y
Note: −4xy and 6xy are added because they are Like Terms.
Also note: 6yx means the same thing as 6xy