$\sqrt{3} x^{2}+10 x+7 \sqrt{3}=0$
$\Rightarrow 3 x^{2}+10 \sqrt{3} x+21=0 \quad$ (Multiplying both sides by $\left.\sqrt{3}\right)$
$\Rightarrow 3 x^{2}+10 \sqrt{3} x=-21$
$\Rightarrow(\sqrt{3} x)^{2}+2 \times \sqrt{3} x \times 5+5^{2}=-21+5^{2} \quad$ (Adding $5^{2}$ on both sides)
$\Rightarrow(\sqrt{3} x+5)^{2}=21+25=4=2^{2}$
$\Rightarrow \sqrt{3} x+5=\pm 2$
$\Rightarrow \sqrt{3} x+5=2$ or $\sqrt{3} x+5=-2$
$\begin{array}{l}
\Rightarrow \sqrt{3} x=-3 \text { or } \sqrt{3} x=-7 \\
\Rightarrow x=-\frac{3}{\sqrt{3}}=-\sqrt{3} \text { or } x=-\frac{7}{\sqrt{3}}=-\frac{7 \sqrt{3}}{3}
\end{array}$

Hence, $-\sqrt{3}$ and $-\frac{7 \sqrt{3}}{3}$ are the roots of the given equation.