Background
What does "factor a quadratic expression" mean? It means to write
the quadratic expression
as the product of two linear expressions. In symbols, to factor
x^{2} + bx + c, means to find the numbers p and q such that p + q
= b
and pq = c.
x^{2} + bx + c = (x + p)(x + q)
Example:
x^{2} + 7x + 10 = (x + 5)(x + 2)
5 + 2 = 7 and (5)(2) = 10
If we have a quadratic expression whose leading coefficient is not 1,
like ax^{2} + bx + c where a ≠ 1, what we need to do in order to
factor it is find the
factors of a and and c and so that the
sum of the outer and inner products (mq and pn) is b.
In symbols,
ax^{2} + bx + c = (mx + p)(nx + q) 
WarmUp 5x^{2}  18x  8 matches
(5x + 2)(x  4).
â€œBecause the last term in the quadratic expression
is negative, the signs of the constant factors will be
different. The factor pair for 5 is 5 and 1. The factor
pairs for 8 are 8 and 1, and 4 and 2. Because 5 times
4 is 20, 2 times 1 is 2, and 20 plus 2 is 18, the
factored expression is (5x + 2)(x  4).â€
