Question

question
factor completely over the integers.
$x^{4}-2x^{2}-15$

Explanation:

Step1: Substitute $u=x^2$

Let $u = x^2$, rewrite the polynomial:
$u^2 - 2u - 15$

Step2: Factor the quadratic in $u$

Find two integers that multiply to $-15$ and add to $-2$: $-5$ and $3$.
$(u - 5)(u + 3)$

Step3: Substitute back $u=x^2$

Replace $u$ with $x^2$:
$(x^2 - 5)(x^2 + 3)$

Step4: Check for further integer factors

$x^2 -5$ and $x^2+3$ cannot be factored over integers.

Answer:

$(x^2 - 5)(x^2 + 3)$