Answer
$(8+10z)(64-80z+100z^2)$
Work Step by Step
$\bf{\text{Solution Outline:}}$
To factor the given expression, $
512+1000z^3
,$ use the factoring of the sum or difference of $2$ cubes.
$\bf{\text{Solution Details:}}$
The expressions $
512
$ and $
1000z^3
$ are both perfect cubes (the cube root is exact). Hence, $
512+1000z^3
$ is a $\text{
sum
}$ of $2$ cubes. Using the factoring of the sum or difference of $2$ cubes which is given by $a^3+b^3=(a+b)(a^2-ab+b^2)$ or by $a^3-b^3=(a-b)(a^2+ab+b^2)$ the expression above is equivalent to
\begin{array}{l}\require{cancel}
(8)^3+(10z)^3
\\\\=
(8+10z)[(8^2)-8(10z)+(10z)^2]
\\\\=
(8+10z)(64-80z+100z^2)
.\end{array}