Question

factor $15t^{4} - 7t^{2} - 2$ completely.

Explanation:

Step1: Substitute $u=t^2$

Let $u = t^2$, so the polynomial becomes:
$15u^2 - 7u - 2$

Step2: Factor quadratic in $u$

Find two numbers that multiply to $15\times(-2)=-30$ and add to $-7$. These numbers are $-10$ and $3$. Rewrite and factor:
$15u^2 -10u +3u -2 = 5u(3u-2)+1(3u-2)=(5u+1)(3u-2)$

Step3: Substitute back $u=t^2$

Replace $u$ with $t^2$:
$(5t^2 +1)(3t^2 -2)$

Answer:

$(5t^2 + 1)(3t^2 - 2)$