To simplify the expression \((9 + \sqrt{-4}) + (-9 - \sqrt{-16})\), we can start by simplifying each part.
First, let's simplify \(\sqrt{-4}\): \[ \sqrt{-4} = \sqrt{4} \cdot \sqrt{-1} = 2i \]
Now, we simplify \(\sqrt{-16}\): \[ \sqrt{-16} = \sqrt{16} \cdot \sqrt{-1} = 4i \]
Now we can substitute these values back into the expression:
\[ (9 + 2i) + (-9 - 4i) \]
Now, combine the real parts and the imaginary parts:
For the real parts: \[ 9 + (-9) = 0 \]
For the imaginary parts: \[ 2i + (-4i) = 2i - 4i = -2i \]
Thus, the simplified expression is: \[ 0 - 2i = -2i \]
So, the final answer is: \[ \boxed{-2i} \]