To solve the equation \(2^{2x + 2} = 2^{3x}\), we can use the property of exponents that states if the bases are the same, we can set the exponents equal to each other:
\[ 2x + 2 = 3x \]
Now, we can isolate \(x\) by rearranging the equation. First, subtract \(2x\) from both sides:
\[ 2 = 3x - 2x \]
This simplifies to:
\[ 2 = x \]
So, the solution is:
\[ x = 2 \]