To solve the equation

\[
-\frac{4}{(x+2)^2} = \frac{x+2}{16} - \frac{4}{(x+2)^2},
\]

we can first simplify the right side. Notice that \(-\frac{4}{(x+2)^2}\) appears on both sides of the equation. We can add \(\frac{4}{(x+2)^2}\) to both sides, which gives:

\[
0 = \frac{x+2}{16}.
\]

To isolate \(x\), we can multiply both sides by 16:

\[
0 = x + 2.
\]

Now, solving for \(x\):

\[
x = -2.
\]

However, we must check if this solution is valid by substituting back into the original equation. The term \((x + 2)^2\) becomes zero when \(x = -2\), which would make the expression undefined. Thus, \(x = -2\) is not a valid solution due to the undefined nature of the original equation.

Therefore, there are no valid solutions to the equation

\[
-\frac{4}{(x+2)^2} = \frac{x+2}{16} - \frac{4}{(x+2)^2}.
\]