Answer
$(5c^2-2d^2)(25c^4+10c^2d^2+4d^4)$
Work Step by Step
Using $a^3+b^3=(a+b)(a^2-ab+b^2)$ or the factoring of 2 cubes, the factored form of the given expression, $
125c^6-8d^6
,$ is
\begin{array}{l}
(5c^2-2d^2)[ (5c^2)^2-(5c^2)(-2d^2)+(-2d^2)^2]
\\\\=
(5c^2-2d^2)(25c^4+10c^2d^2+4d^4)
.\end{array}