Answer
$x=-\frac{2}{3}$
Work Step by Step
$(625)^{x}=(\frac{1}{25})^{x+2}$
Rewriting $625$ as $25^{2}$ and $\frac{1}{25}$ as $25^{-1}$, we have
$(25^{2})^{x}=(25^{-1})^{x+2}$
$\implies 25^{2x}=25^{-x-2}$
$\implies 2x=-x-2$
$\implies 2x+x=-2$ or $3x=-2$
$\implies x=-\frac{2}{3}$
Check:
$(625)^{-\frac{2}{3}}=(25^2)^{-\frac{2}{3}}=\left(\frac{1}{25}\right)^{\frac{4}{3}}$
$\left(\frac{1}{25}\right)^{(-\frac{2}{3}+2)}=\left(\frac{1}{25}\right)^{\frac{4}{3}}$