Question

question
factor completely over the integers.
$x^{4}+7x^{2}+6$

Explanation:

Step1: Substitute $u=x^2$

Let $u=x^2$, so the polynomial becomes:
$u^2 + 7u + 6$

Step2: Factor the quadratic in $u$

Find two integers that multiply to 6 and add to 7 (1 and 6):
$(u + 1)(u + 6)$

Step3: Substitute back $u=x^2$

Replace $u$ with $x^2$:
$(x^2 + 1)(x^2 + 6)$

Step4: Check for further factorization

Over integers, $x^2+1$ and $x^2+6$ cannot be factored, as they have no integer roots.

Answer:

$(x^2 + 1)(x^2 + 6)$