1. **State the problem:** Solve the equation $$2^{-2x+5} \times \frac{1}{4} = 2^{x+2}$$ for $x$. 2. **Rewrite the equation using properties of exponents:** Note that $\frac{1}{4} = 2^{-2}$, so the equation becomes: $$2^{-2x+5} \times 2^{-2} = 2^{x+2}$$ 3. **Combine the left side exponents:** Using $a^m \times a^n = a^{m+n}$, $$2^{-2x+5-2} = 2^{x+2}$$ which simplifies to $$2^{-2x+3} = 2^{x+2}$$ 4. **Set the exponents equal:** Since the bases are the same and nonzero, $$-2x + 3 = x + 2$$ 5. **Solve for $x$:** $$-2x + 3 = x + 2$$ $$3 - 2 = x + 2x$$ $$1 = 3x$$ $$x = \frac{1}{3}$$ **Final answer:** $$x = \frac{1}{3}$$