In trinomial factoring, we are looking for two binomials that multiply to get the given trinomial.
Most trinomials have the form ax² + bx + c.

Basic steps for trinomial factoring:

1. If the trinomial has the same factor in all its terms, find and factor out this GCF.
2. Find the factors of the coefficient of the first term, a. These factors help make up the first term of each binomial.
   This is the reverse of F in the FOIL method.
3. Find the factors of the last term, c. These factors make up the last term of each binomial.
   This is the reverse of L in the FOIL method.
4. Guess and check to find the right combination of factors.

Example 1: Simple trinomial with GCF

^- 7x³ - 7x² + 84x

Step 1.   Find the GCF, if the trinomial has one.
Each term in the trinomial has the factor -7x so let's factor it out.

-7x(x² + x - 12)

Now we will factor the trinomial x² + x - 12.

Step 2.   Find the factors of the coefficient of ax².
The coefficient of x² is 1.

This means x² has only the factors x and x. The equation looks like
-7x(x ... )(x ... )

Step 3.   Find the factors of -12.
Factors of -12: 1, 2, 3, 4, 6, 12, -1, -2, -3, -4, -6, and -12.

Since c is negative, the factors in the last terms have opposite signs. So the equation looks like
-7x(x + ... )(x - ... )

Step 4.   Guess and check combinations of factors.
One way to do this is to put the factors of the coefficients into two columns such that the first column is for a and the second column is for c.
For this example, a = 1 and c = -12. In this case, we have the following combinations:

1 1	-1 12			1 1	1 -12			1 1	-2 6			1 1	2 -6			1 1	-3 4			1 1	3 -4
Try each combination by multiplying across.
1 1		-1 12	-1  12	1 1		1 -12	1  -12	1 1		-2 6	-2  6	1 1		2 -6	2  -6	1 1		-3 4	-3  4	1 1		3 -4	3  -4
Add the products. Which combination equals b?
-1 + 12 = 11				1 + (-12) = -11				-2 + 6 = 4				2 + (-6) = -4				-3 + 4 = 1				3 + (-4) = -1
For the trinomial x2 + x - 12, b = 1.
1 1	-3 4	is the combination that adds up to 1.

This means that x² + x - 12 = (x -3)(x + 4)
The answer is .

Example 2: Trinomial where a ≠ 1

4x² - 16x + 15

Step 1.   Find the GCF, if the trinomial has one.
There is no GCF for this trinomial so we cannot factor anything out.

Step 2.   Find the factors of the coefficient of ax².
The coefficient of 4x² is 4.
Factors of 4: 1, 2, and 4.

The equation looks like either
(x ... )(4x ... ) or (2x ... )(2x ... )

Step 3.   Find the factors of 15.

Since c is positive and b is negative, the factors in the last terms have - signs. So the equation looks like either
(x - ... )(4x - ... ) or (2x - ... )(2x - ... )
So we will look at the negative factors of 15: -1, -3, -5, and -15.

Step 4.   Guess and check combinations of factors.
One way to do this is to put the factors of the coefficients into two columns such that the first column is for a and the second column is for c.
For this example, a = 4 and c = 15. In this case, we have the following combinations:

1 4	-1 -15			1 4	-3 -5			2 2	-1 -15			2 2	-3 -5
Try each combination by multiplying across.
1 4		-1 -15	-4  -15	1 4		-3 -5	-12  -5	2 2		-1 -15	-2  -30	2 2		-3 -5	-6  -10
Add the products. Which combination equals b?
-4 + (-15) = -19				-12 + (-5) = -17				-2 + (-30) = -32				-6 + (-10) = -16
For the trinomial 4x2 - 16x + 15, b = -16.
2 2	-3 -5	is the combination that adds up to -16.

This means that 4x² - 16x + 15 = (2x - 3)(2x - 5)
The answer is .