To solve the equation \( 2^{2x + 2} = 2^{3x} \), we can start by using the property that if the bases are the same, we can set the exponents equal to each other.
So, we equate the exponents:
\[ 2x + 2 = 3x \]
Now, we'll solve for \( x \). First, we can rearrange the equation by subtracting \( 2x \) from both sides:
\[ 2 = 3x - 2x \]
This simplifies to:
\[ 2 = x \]
Thus, the solution for \( x \) is:
\[ \boxed{2} \]