There are numerous ways to set up the multiplication of two binomials.
The first three methods shown here work for multiplying ALL polynomials,
not just binomials. All methods, of
course, give the same answer. 

Multiply (x + 3)(x + 2) 

1.
"Distributive" Method: 
The most universal
method. Applies to all polynomial multiplications, not just to
binomials. 
Start with the first term in the first binomial
 the circled blue X.
Multiply (distribute) this term times EACH of the terms in the second
binomial.
Now, take the second term in the first binomial
 the circled red +3
(notice we take the sign also).
Multiply this term times EACH of the terms in the second
binomial. 
Add the results: x•x + x•2
+ 3•x +3•2
x² + 2x +3x +
6
x² + 5x + 6 Answer
Do you
see the "distributive property" at work?
(x + 3)(x +
2) = x(x + 2) + 3(x + 2)


Before we move on to the next
set up method, let's look at an example of the "distributive" method
involving negative values.

"Distributive" Method: 
Dealing with
negative values. 
Notice how the negative sign is treated as part
of the term following the sign.

Add the results:
x•x + x•(5) +(4)•x +(4)•(5)
x²  5x  4x + 20
x²  9x + 20 Answer 


2.
"Vertical" Method:

This is a vertical
"picture" of the distributive method.
This style applies to all polynomial multiplications. 
x + 2


x + 3




x² + 2x 
multiply "x" from bottom term times "x+2" 
Be sure to line up
the like terms. 
3x + 6 
multiply "3" from bottom term times "x+2" 


x² + 5x + 6 
add the like terms 


3. "Grid" Method 
This is a "table"
version of the distributive method.
This style applies to all polynomial multiplications. 
To multiply by
the grid method, place one binomial
at the top of a 2x2 grid (for binomials) and the
second binomial on the side of the grid. Place the terms such that
each term with its sign lines up with a row or column of the grid.
Multiply the rows and columns of the
grid to complete the interior of the grid. Finish by adding together
the entries inside the grid.
2x + 6 + (x)(x) + 3x
2x + 6 + x^{2} + 3x
Answer:
x^{2} + 5x + 6

C A U T I O N !!!
There are set up methods that work ONLY
for binomials. While these set ups may be helpful to understanding
binomial multiplication, you must remember that they do not extend to
other types of multiplications, such as a binomial times a trinomial.
You will have to go back to the "distributive method" for
these other polynomial
multiplications. 

For Binomial
Multiplication ONLY!
"FOIL" Method:
multiply First
Outer
Inner
Last
The words/letters used to describe the FOIL process
pertain to the distributive method for multiplying two binomials.
These words/letters do not apply to other multiplications such as a binomial times a trinomial.

"Algebra Tile" Method 
While this method is helpful for
understanding how binomials are multiplied, it is not easily applied to
ALL multiplications and may not be practical
for overall use. 
The example shown here is
for binomial multiplication only!
To multiply binomials using algebra tiles, place one
expression at the top of the grid and the second
expression on the
side of the grid. You MUST
maintain straight lines when you are filling in the center of the grid. The tiles
needed to complete the inner grid will be your answer.
Answer:
x² + 5x
+ 6

