1. **State the problem:** Solve for $x$ in the equation $$2^{2x+2} = 2^{3x}$$. 2. **Use the property of exponents:** If $a^m = a^n$ and $a > 0$, $a \neq 1$, then $m = n$. 3. **Apply this property:** Since the bases are the same (base 2), set the exponents equal: $$2x + 2 = 3x$$ 4. **Solve for $x$:** $$2x + 2 = 3x$$ Subtract $2x$ from both sides: $$\cancel{2x} + 2 = \cancel{2x} + 3x - 2x$$ $$2 = x$$ 5. **Final answer:** $$x = 2$$