Question

find the binomial that completes the factorization.
$2s^{3}t + 128t = 2t(\square)(s^{2} - 4s + 16)$

Explanation:

Step1: Factor out common term

First, factor $2t$ from the left-hand side:
$2s^3t + 128t = 2t(s^3 + 64)$

Step2: Recognize sum of cubes

Note that $s^3 + 64$ is a sum of cubes, where $s^3 + 4^3$. The sum of cubes formula is $a^3 + b^3 = (a + b)(a^2 - ab + b^2)$. Here $a=s$, $b=4$:
$s^3 + 64 = (s + 4)(s^2 - 4s + 16)$

Step3: Substitute back into expression

Replace $s^3 + 64$ in the factored form from Step1:
$2t(s^3 + 64) = 2t(s + 4)(s^2 - 4s + 16)$

Answer:

$s + 4$