Question

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

Explanation:

Step1: Substitute $u=x^2$

Let $u = x^2$, rewrite the polynomial:
$u^2 + 6u - 7$

Step2: Factor the quadratic in $u$

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

Step3: Substitute back $u=x^2$

Replace $u$ with $x^2$:
$(x^2 + 7)(x^2 - 1)$

Step4: Factor difference of squares

$x^2 - 1$ is a difference of squares, so:
$(x^2 + 7)(x + 1)(x - 1)$

Answer:

$(x - 1)(x + 1)(x^2 + 7)$