1. Objective
Factor quadratic trinomials of the form ax2 + bx + c.

2. When you multiply (3x + 2)(2x + 5), the coefficient of the x2-term is the product of the coefficients of the x-terms. Also, the constant term in the trinomial is the product of the constants in the binomials.
(3x + 2)(2x + 5) = 6x2 + 19x + 10

3. Example 1: Factoring ax2 + bx + c by Guess and Check
Factor 6x2 + 11x + 4 by guess and check.
( )( )
Write two sets of parentheses.
( x + )( x + )
The first term is 6x2, so at least one variable term has a coefficient other than 1.
The coefficient of the x2 term is 6. The constant term in the trinomial is 4.
(2x + 4)(3x + 1) = 6x2 + 14x + 4 
(1x + 4)(6x + 1) = 6x2 + 25x + 4 
Try factors of 6 for the coefficients and factors of 4 for the constant terms.
(1x + 2)(6x + 2) = 6x2 + 14x + 4 
(1x + 1)(6x + 4) = 6x2 + 10x + 4 
(3x + 4)(2x + 1) = 6x2 + 11x + 4 

4.     Check It Out! Example 1a
Factor each trinomial by guess and check.
6x2 + 11x + 3
( )( )
Write two sets of parentheses.
( x + )( x + )
The first term is 6x2, so at least one variable term has a coefficient other than 1.
The coefficient of the x2 term is 6. The constant term in the trinomial is 3.
(2x + 1)(3x + 3) = 6x2 + 9x + 3 
Try factors of 6 for the coefficients and factors of 3 for the constant terms.
(1x + 3)(6x + 1) = 6x2 + 19x + 3 
(1x + 1)(6x + 3) = 6x2 + 9x + 3 
(3x + 1)(2x + 3) = 6x2 + 11x + 3 

5.     Check It Out! Example 1b
Factor each trinomial by guess and check.
3x2 – 2x – 8
( )( )
Write two sets of parentheses.
The first term is 3x2, so at least one variable term has a coefficient other than 1.
( x + )( x + )
The coefficient of the x2 term is 3. The constant term in the trinomial is –8.
(1x – 1)(3x + 8) = 3x2 + 5x – 8 
Try factors of 3 for the coefficients and factors of 8 for the constant terms. 
(1x – 4)(3x + 2) = 3x2 – 10x – 8
(1x – 8)(3x + 1) = 3x2 – 23x – 8 
(1x – 2)(3x + 4) = 3x2 – 2x – 8 

6. Example 3A: Factoring ax2 + bx + c When c is Negative
Factor each trinomial. Check your answer.
3n2 + 11n – 4
a = 3 and c = – 4,
Outer + Inner = 11 .
( n + )( n+ )
Factors of 3 Factors of 4 Outer + Inner
1 and 3
–1 and 4
1(4) + 3(–1) = 1
–2 and 2
1(2) + 3(–2) = – 4
–4 and 1
1(1) + 3(–4) = –11 
4 and –1
1(–1) + 3(4) = 11 
(n + 4)(3n – 1)
Use the Foil method.
Check (n + 4)(3n – 1) = 3n2 – n + 12n – 4
= 3n2 + 11n – 4 

7. Example 3C: Factoring ax2 + bx + c When c is Negative
Factor each trinomial. Check your answer.
4x2 – 15x – 4
a = 4 and c = –4,
Outer + Inner = –15.
( x + )( x+ )
Factors of 4 Factors of – 4 Outer + Inner
1 and 4
–1 and 4
1(4) – 1(4) = 0
–2 and 2
1(2) – 2(4) = –6
–4 and 1
1(1) – 4(4) = –15  
(x – 4)(4x + 1)
Use the Foil method.
Check (x – 4)(4x + 1) = 4x2 + x – 16x – 4
= 4x2 – 15x – 4 

8. Example 4A: Factoring ax2 + bx + c When a is Negative
Factor –2x2 – 5x – 3.
–1(2x2 + 5x + 3)
Factor out –1.
a = 2 and c = 3;
Outer + Inner = 5
–1( x + )( x+ )
Factors of 2 Factors of Outer + Inner
1 and 2
3 and 1
1(1) + 3(2) = 7  
1 and 3
1(3) + 1(2) = 5
(x + 1)(2x + 3)
–1(x + 1)(2x + 3)