Question

factor completely over the integers.\\(x^4 + 10x^2 + 24\\)

Explanation:

Step1: Substitute $u=x^2$

Let $u = x^2$, so the polynomial becomes:
$u^2 + 10u + 24$

Step2: Factor the quadratic in $u$

Find two integers that multiply to 24 and add to 10 (6 and 4):
$u^2 + 10u + 24 = (u + 6)(u + 4)$

Step3: Substitute back $u=x^2$

Replace $u$ with $x^2$:
$(x^2 + 6)(x^2 + 4)$

Step4: Check for further integer factorization

$x^2 + 6$ and $x^2 + 4$ cannot be factored over integers, as they have no integer roots.

Answer:

$(x^2 + 4)(x^2 + 6)$