jmc

Rewriting both sides with $5$ as the base, we have ${\left(\frac{1}{25}\right)}^{x+2}=(5^{-2}{)}^{x+2}=5^{-2x-4}$, and $125^{-x}=(5^3{)}^{-x}=5^{-3x}$, which means our equation is: $$5^{-2x-4}=5^{-3x}.$$Then, by setting the exponents equal to each other, we obtain $$-2x-4=-3x.$$This yields our solution $\boxed{x=4}$