$\begin{array}{l}
4 x^{2}+4 \sqrt{3} x+3=0 \\
\Rightarrow 4 x^{2}+4 \sqrt{3} x=-3 \\
\Rightarrow(2 x)^{2}+2 \times 2 x \times \sqrt{3}+(\sqrt{3})^{2}=-3+(\sqrt{3})^{2}
\end{array}$
$\begin{array}{l}
\Rightarrow(2 x+\sqrt{3})^{2}=-3+3=0 \\
\Rightarrow 2 x+\sqrt{3}=0 \\
\Rightarrow x=-\frac{\sqrt{3}}{2}
\end{array}$

Hence, $-\frac{\sqrt{3}}{2}$ is the repeated root of the given equation.