Answer
$(5y^2+z)(25y^4-5y^2z+z^2)$
Work Step by Step
$\bf{\text{Solution Outline:}}$
To factor the given expression, $
125y^6+z^3
,$ use the factoring of the sum/difference of $2$ cubes.
$\bf{\text{Solution Details:}}$
Using the factoring of the sum/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}
(5y^2+z)[(5y^2)^2-5y^2(z)+(z)^2]
\\\\=
(5y^2+z)(25y^4-5y^2z+z^2)
.\end{array}