A Negative Product

To factor $x^{2}+2x-8$, find two numbers with product -8 and sum 2. A negative product means the two numbers have opposite signs. One number is positive and the other is negative.

List the factor pairs of 8, which are 1 and 8, and 2 and 4. Put a minus sign on one number of each pair and look for the pair that adds to 2. The pair 4 and -2 works, because $4-2 = 2$. So $x^{2}+2x-8 = \left(x+4\right)\left(x-2\right)$. The factor $x-2$ means $x-2$, written in the usual short way.

For $x^{2}+bx+c$, the two numbers in the factors must multiply to $c$ and add to $b$. If $c$ is negative, the two numbers have opposite signs. If $c$ is positive, they have the same sign. Then use $b$ to decide which number gets which sign.