Question

factor $x^4 + 11x^2 + 18$ completely. all factors in your answer should have integer coefficients.

Explanation:

Step1: Substitute $u=x^2$

Let $u = x^2$, rewrite the polynomial:
$u^2 + 11u + 18$

Step2: Factor the quadratic in $u$

Find two numbers that multiply to 18 and add to 11 (2 and 9):
$(u + 2)(u + 9)$

Step3: Substitute back $u=x^2$

Replace $u$ with $x^2$:
$(x^2 + 2)(x^2 + 9)$

Step4: Check for further factoring

$x^2+2$ and $x^2+9$ cannot be factored into linear terms with integer coefficients.

Answer:

$(x^2 + 2)(x^2 + 9)$