To simplify the expression \(-4\sqrt{3} + 4\sqrt{108}\), let's first simplify \(\sqrt{108}\).
We start by breaking down 108:
\[ 108 = 36 \times 3 = 6^2 \times 3 \]
Thus, we can simplify:
\[ \sqrt{108} = \sqrt{36 \times 3} = \sqrt{36} \cdot \sqrt{3} = 6\sqrt{3} \]
Now we can substitute \(\sqrt{108}\) back into the original expression:
\[ -4\sqrt{3} + 4\sqrt{108} = -4\sqrt{3} + 4(6\sqrt{3}) \]
Now multiply \(4\) by \(6\sqrt{3}\):
\[ 4(6\sqrt{3}) = 24\sqrt{3} \]
Now the expression becomes:
\[ -4\sqrt{3} + 24\sqrt{3} \]
Now combine like terms:
\[ (-4 + 24)\sqrt{3} = 20\sqrt{3} \]
Thus, the simplified expression is:
\[ \boxed{20\sqrt{3}} \]