The
AC Method for Factoring Trinomials. 
Factor: 
1. Multiply
the leading coefficient (the number in front of x^{2}
) times the constant term (the last term). Remove the
leading coefficient.
(Obviously an illegal algebraic move!) 

2. Factor this
new trinomial. 

3. Replace
x with the original leading coefficient times x. In this example, the leading coefficient
is 2. 

4. Factor. 

5. Divide by
the original leading coefficient. 


How can an "illegal" move
result in a correct answer? Is this
really a valid method of factoring?
And if so, why does it work? 
Why it works:
The secret to understanding this
method is to realize that these steps are actually a shorthand for a
more complex process of multiplication and replacement.
Let's take one more look at the whole process. The
AC Method is simply a shorthand version of the
following procedure.
Multiply the entire
expression times the leading coefficient: 

Distribute: 

Rewrite: 

Replace 2x
with another variable, such as m,
where 2x = m: 

Factor: 

Replace m
with 2x: 

Factor: 

These factors now equal
what we started with, which is TWICE the original problem:
=

Divide both sides
by 2:
= 