$\begin{array}{l}
\sqrt{3} x^{2}+11 x+6 \sqrt{3}=0 \\
\Rightarrow \sqrt{3} x^{2}+9 x+2 x+6 \sqrt{3}=0 \\
\Rightarrow \sqrt{3} x(x+3 \sqrt{3})+2(x+3 \sqrt{3})=0
\end{array}$
$\begin{array}{l}
\Rightarrow(x+3 \sqrt{3})(\sqrt{3} x+2)=0 \\
\Rightarrow x+3 \sqrt{3}=0 \text { or } \sqrt{3} x+2=0 \\
\Rightarrow x=-3 \sqrt{3} \text { or } x=\frac{-2}{\sqrt{3}}=\frac{-2 \times \sqrt{3}}{\sqrt{3} \times \sqrt{3}}=\frac{-2 \sqrt{3}}{3}
\end{array}$

Hence, the roots of the equation are $-3 \sqrt{3}$ and $\frac{-2 \sqrt{3}}{3}$.