First, find the value of $f(x)$ when $x=-3$ \[ \begin{aligned} f(-3) & =2^{(-3+1)+4} \\ & =2^{-2}+4 \\ & = \\ & =12.4 \end{aligned} \]

Step 1 ：First, find the value of \(f(x)\) when \(x=-3\)

Step 2 ：Substitute \(x=-3\) into the function \(f(x) = 2^{x+1} + 4\)

Step 3 ：Calculate the result: \(f(-3) = 2^{-2} + 4 = 4.25\)

Step 4 ：Final Answer: The value of \(f(-3)\) is \(\boxed{4.25}\)