Find the Factorization: x^2 + 12x + 32 as a Product

To solve the problem of factoring the quadratic expression $f(x) = x^2 + 12x + 32$, we will follow these steps:

• Identify the pairs of integers whose product is 32.
• Find the pair that also adds up to 12.
• Use this pair to express the quadratic in its factored form.

Let's proceed with these steps:

Step 1: List all pairs of integers that multiply to 32:
1 and 32
2 and 16
4 and 8

Step 2: Determine which pair of these adds up to 12:
Checking each:
$1 + 32 = 33$
$2 + 16 = 18$
$4 + 8 = 12$

The pair 4 and 8 adds up to 12.

Step 3: Use this pair to factor the quadratic expression:
Thus, $f(x) = (x + 4)(x + 8)$.

Therefore, the factored form of $f(x) = x^2 + 12x + 32$ is $(x + 8)(x + 4)$.