When we rewrite expressions in factored form, it is helpful to remember that:

• Multiplying two positive numbers or two negative numbers results in a positive product.
• Multiplying a positive number and a negative number results in a negative product.

This means that if we want to find two factors whose product is 10, the factors must both be positive or both be negative. If we want to find two factors whose product is -10, one of the factors must be positive and the other negative.

Suppose we wanted to rewrite $x^2 -8x + 7$ in factored form. Recall that subtracting a number can be thought of as adding the opposite of that number, so that expression can also be written as $x^2 + \text-8x + 7$. We are looking for two numbers that:

• Have a product of 7. The candidates are 7 and 1, and -7 and -1.
• Have a sum of -8. Only -7 and -1 from the list of candidates meet this condition.

The factored form of $x^2 -8x + 7$ is therefore $(x + \text-7)(x + \text-1)$ or, written another way, $(x-7)(x-1)$.

To write $x^2 + 6x - 7$ in factored form, we would need two numbers that:

• Multiply to make -7. The candidates are 7 and -1, and -7 and 1.
• Add up to 6. Only 7 and -1 from the list of candidates add up to 6.

The factored form of $x^2 + 6x - 7$ is $(x+7)(x-1)$.