Factoring ax^{2} + bx + c
This section explains how to factor expressions of the
form ax^{2} + bx + c, where a, b, and c are integers.
First, factor out all constants which evenly divide all three terms. If
a is negative, factor out 1. This will leave an expression of the
form d (ax^{2} + bx + c), where a, b, c, and d are integers, and
a > 0. We can now turn to factoring the inside expression.
Here is how to factor an expression ax^{2} + bx + c, where a > 0:
 Write out all the pairs of numbers that, when multiplied, produce
a.
 Write out all the pairs of numbers that, when multiplied, produce
c.
 Pick one of the a pairs  (a_{1}, a_{2})  and one of the c
pairs  (c_{1}, c_{2}).
 If c > 0: Compute a_{1}c_{1} + a_{2}c_{2}. If  a_{1}c_{1} + a_{2}c_{2} = b,
then the factored form of the quadratic is

(a_{1}x + c_{2})(a_{2}x + c_{1}) if b > 0.

(a_{1}x  c_{2})(a_{2}x  c_{1}) if b < 0.
 If a_{1}c_{1} + a_{2}c_{2}≠b, compute a_{1}c_{2} + a_{2}c_{1}. If a_{1}c_{2} + a_{2}c_{1} = b, then the factored form of the quadratic is (a_{1}x + c_{1})(a_{2}x + c_{2}) or (a_{1}x + c_{1})(a_{2}x + c_{2}). If a_{1}c_{2} + a_{2}c_{1}≠b, pick another set of pairs.
 If c < 0: Compute a_{1}c_{1} a_{2}c_{2}. If  a_{1}c_{1}  a_{2}c_{2} = b, then
the factored form of the quadratic is:
(a_{1}x  c_{2})(a_{2}x + c_{1}) where a_{1}c_{1} > a_{2}c_{2} if b > 0 and
a_{1}c_{1} < a_{2}c_{2} if b < 0.
Using FOIL, the outside pair plus (or minus) the inside pair must equal
b.
 Check.
Example 1: Factor 3x^{2}  8x + 4.
 Numbers that produce 3: (1, 3).
 Numbers that produce 4: (1, 4), (2, 2).

 (1, 3) and (1, 4): 1(1) + 3(4) = 11≠8. 1(4) + 3(1) = 7≠ = 8.
 (1, 3) and (2, 2): 1(2) + 3(2) = 8.

(x  2)(3x  2).
 Check: (x  2)(3x  2) = 3x^{2} 2x  6x + 4 = 3x^{2}  8x + 4.
Example 2: Factor 12x^{2} + 17x + 6.
 Numbers that produce 12: (1, 12), (2, 6), (3, 4).
 Numbers that produce 6: (1, 6), (2, 3).

 (1, 12) and (1, 6): 1(1) + 12(6) = 72. 1(6) + 12(1) = 18.
 (1, 12) and (2, 3): 1(2) + 12(3) = 38. 1(3) + 12(2) = 27.
 (2, 6) and (1, 6): 2(1) + 6(6) = 38. 2(6) + 6(1) = 18.
 (2, 6) and (2, 3): 2(2) + 6(3) = 22. 2(3) + 6(2) = 18.
 (3, 4) and (1, 6): 3(1) + 4(6) = 27. 3(6) + 4(1) = 22.
 (3, 4) and (2, 3): 3(2) + 4(3) = 18. 3(3) + 4(2) = 17.
(3x + 2)(4x + 3).
 Check: (3x + 2)(4x + 3) = 12x^{2} +9x + 8x + 6 = 12x^{2} + 17x + 6.
Example 3: Factor 4x^{2}  5x  21.
 Numbers that produce 4: (1, 4), (2, 2).
 Numbers that produce 21: (1, 21), (3, 7).

 (1, 4) and (1, 21): 1(1) 4(21) =  83. 1(21)  4(1) = 17.
 (1, 4) and (3, 7): 1(3)  4(7) =  25. 1(7)  4(3) =  5.
(x  3)(4x + 7).
 Check: (x  3)(4x + 7) = 4x^{2} +7x  12x  21 = 4x^{2}  5x  21.