The Sign Map

There are four sign patterns, and one sign rule covers them all. Look at the sign of $c$ first. If $c$ is positive, the two numbers have the same sign, and the sign of $b$ says which sign it is. For $x^{2}+7x+6$ the numbers are +1 and +6. For $x^{2}-5x+6$ the numbers are -2 and -3.

If $c$ is negative, the two numbers have opposite signs. Find the pair by its difference, and give the larger number the sign of $b$. For $x^{2}+x-12$ the numbers are +4 and -3. For $x^{2}-x-12$ the numbers are -4 and +3.

Use the signs of $b$ and $c$ to decide the signs of the two numbers. Then check the factor pairs to find the pair whose product is $c$ and whose sum is $b$.