Question

what is the factored form of this expression?
$x^2 + 9x + 16$
a. $(x + 3)^2$
b. the expression cannot be factored.
c. $(x - 4)(x + 4)$
d. $(x + 4)^2$

Explanation:

Step1: Test Option A

Expand $(x+3)^2$:
$$(x+3)^2 = x^2 + 6x + 9$$
This does not match $x^2 + 9x + 16$.

Step2: Test Option C

Expand $(x-4)(x+4)$:
$$(x-4)(x+4) = x^2 - 16$$
This does not match $x^2 + 9x + 16$.

Step3: Test Option D

Expand $(x+4)^2$:
$$(x+4)^2 = x^2 + 8x + 16$$
This does not match $x^2 + 9x + 16$.

Step4: Check for integer factorization

Find two integers that multiply to $16$ and add to $9$. The factor pairs of $16$ are $(1,16), (2,8), (4,4)$. Their sums are $17, 10, 8$—none equal $9$. No integer factorization exists.

Answer:

B. The expression cannot be factored.