Question

question
factor completely over the integers.
$x^{4}+3x^{2}-40$

Explanation:

Step1: Substitute $u=x^2$

Let $u = x^2$, so the polynomial becomes:
$u^2 + 3u - 40$

Step2: Factor quadratic in $u$

Find two integers that multiply to $-40$ and add to $3$ (which are $8$ and $-5$):
$(u + 8)(u - 5)$

Step3: Substitute back $u=x^2$

Replace $u$ with $x^2$:
$(x^2 + 8)(x^2 - 5)$

Step4: Factor difference of squares

$x^2 - 5$ cannot be factored over integers, so stop here.

Answer:

$(x^2 + 8)(x^2 - 5)$