Answer
$ 5(2p-5q)^{2}$
Work Step by Step
Factor out the greatest common factor, $5$
$20p^{2}-100pq+125q^{2}=5(4p^{2}-20pq+25q^{2})=*$
Test whether the parentheses hold a perfect square,
$(A-B)^{2}=A^{2}-2AB+B^{2}$
First term: $A^{2}=(2p)^{2}\Rightarrow A=2p$
Third term: $B^{2}=(5q)^{2}\Rightarrow B=5q$
Test:$\qquad$ does $-2AB$ equal the middle term?
$-2AB=-2(2p)5q)=-20pq\qquad$ ... yes, it does.
$\Rightarrow$The parentheses hold a perfect square, $(2p-5q)^{2}$
$*...= 5(2p-5q)^{2}$
