Question

factor the trinomial.
$x^2 + 4x - 77$

Explanation:

Step 1: Find two numbers

We need two numbers that multiply to \(-77\) (the constant term) and add up to \(4\) (the coefficient of the middle term). Let's list the factor pairs of \(77\): \(1\) and \(77\), \(7\) and \(11\). Since the product is negative, one number is positive and the other is negative. We want the pair where the positive number is larger than the negative one so that their sum is positive. Let's check \(-7\) and \(11\): \(-7\times11 = -77\) and \(-7 + 11=4\).

Step 2: Factor the trinomial

Using the two numbers we found, we can factor the trinomial \(x^{2}+4x - 77\) as \((x - 7)(x + 11)\). We check by multiplying \((x - 7)(x + 11)=x^{2}+11x-7x - 77=x^{2}+4x - 77\), which matches the original trinomial.

Answer:

\((x - 7)(x + 11)\)