Question

factor $x^4 + 5x^2 + 6$ completely.
all factors in your answer should have integer coefficients.

Explanation:

Step1: Substitute $u=x^2$

Let $u = x^2$, so the polynomial becomes:
$u^2 + 5u + 6$

Step2: Factor the quadratic in $u$

Find two integers that multiply to 6 and add to 5 (2 and 3):
$u^2 + 5u + 6 = (u + 2)(u + 3)$

Step3: Substitute back $u=x^2$

Replace $u$ with $x^2$ to get factors in terms of $x$:
$(x^2 + 2)(x^2 + 3)$

Answer:

$(x^2 + 2)(x^2 + 3)$