$4\sin(x) + 2\sqrt{3} = 0 \Rightarrow \sin(x) = -\frac{\sqrt{3}}{2} $
$\Rightarrow \sin\left(-\frac{\pi}{3}\right)$
$\Rightarrow \begin{cases}
x = 2k\pi - \frac{\pi}{3} \\
x = 2k\pi + \frac{4\pi}{3}
\end{cases} \quad (k \in \mathbb{Z})$
$ \begin{cases}
x = \frac{5\pi}{3} \\
x = \frac{4\pi}{3}
\end{cases}$