We need to evaluate the integral $\int 2^{x+2}dx$.

First, we can rewrite the integrand as $2^{x+2} = 2^x \cdot 2^2 = 4 \cdot 2^x$.

So, the integral becomes $\int 4 \cdot 2^x dx = 4 \int 2^x dx$.

The integral of $a^x$ is $\frac{a^x}{\log a} + C$. Therefore, $\int 2^x dx = \frac{2^x}{\log 2} + C$.

Thus, $4 \int 2^x dx = 4 \cdot \frac{2^x}{\log 2} + C = \frac{4 \cdot 2^x}{\log 2} + C = \frac{2^2 \cdot 2^x}{\log 2} + C = \frac{2^{x+2}}{\log 2} + C$.