Answer
$(4b^2+25c^2)(2b+5c)(2b-5c)$
Work Step by Step
$\bf{\text{Solution Outline:}}$
To factor the given expression, $
16b^4-625c^4
,$ use the factoring of the difference of $2$ squares.
$\bf{\text{Solution Details:}}$
The expressions $
16b^4
$ and $
625c^4
$ are both perfect squares (the square root is exact) and are separated by a minus sign. Hence, $
16b^4-625c^4
,$ is a difference of $2$ squares. Using the factoring of the difference of $2$ squares, which is given by $a^2-b^2=(a+b)(a-b),$ the expression above is equivalent to
\begin{array}{l}\require{cancel}
(4b^2)^2-(25c^2)^2
\\\\=
(4b^2+25c^2)(4b^2-25c^2)
.\end{array}
The expressions $
4b^2
$ and $
25c^2
$ are both perfect squares (the square root is exact) and are separated by a minus sign. Hence, $
4b^2-25c^2
,$ is a difference of $2$ squares. Using the factoring of the difference of $2$ squares, which is given by $a^2-b^2=(a+b)(a-b),$ the expression above is equivalent to
\begin{array}{l}\require{cancel}
(4b^2+25c^2)[(2b)^2-(5c)^2]
\\\\=
(4b^2+25c^2)(2b+5c)(2b-5c)
.\end{array}
