How do you factor the expression x^2 + 12x + 32?

To factor the quadratic expression x² + 12x + 32, we need to find two numbers that multiply to the constant term (32) and add up to the linear coefficient (12).

We can list the pairs of factors of 32:

• 1 and 32
• 2 and 16
• 4 and 8

Among these pairs, the numbers 4 and 8 add up to 12. Hence, we can rewrite the quadratic expression using these factors:

x² + 4x + 8x + 32

Now, we can group the terms:

(x² + 4x) + (8x + 32)

Next, we factor out the common factors from each group:

x(x + 4) + 8(x + 4)

Now, we can factor out the common binomial factor (x + 4):

(x + 4)(x + 8)

To summarize, the factored form of the expression x² + 12x + 32 is:

(x + 4)(x + 8).