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

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Answer

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

Steps

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 $\boxed{4.25}$