Question

question
factor completely over the integers.
$x^4 - 14x^2 + 48$

Explanation:

Step1: Substitute $u=x^2$

Let $u = x^2$, so the expression becomes:
$u^2 - 14u + 48$

Step2: Factor the quadratic in $u$

Find two integers that multiply to 48 and add to -14: -6 and -8.
$(u - 6)(u - 8)$

Step3: Substitute back $u=x^2$

Replace $u$ with $x^2$:
$(x^2 - 6)(x^2 - 8)$

Step4: Factor difference of squares

Note $x^2 - 6$ and $x^2 - 8$ cannot be factored over integers, so we stop here.

Answer:

$(x^2 - 6)(x^2 - 8)$