$\begin{equation*}\begin{array}{l}
2^{n+2} + 2n + 1 = 2^{n+3} - n - 4 = 2 \cdot 2^{n+2} - n - 4 \\
2 \cdot 2^{n+2} - 2^{n+2} = n + 4 + 2n + 1 = 0 \\
2^{n+2} = 3n + 5
\end{array}\end{equation*}$