$2cos3x+\sqrt3sinx+cosx=0$$\Leftrightarrow cos3x+(\frac{\sqrt3}{2}sinx+\frac{1}{2}cosx)=0$
$\Leftrightarrow cos3x+(sinx.sin\frac{\pi}{3}+cosx.cos\frac{\pi}{3})=0$
$\Leftrightarrow cos3x+cos(x-\frac{\pi}{3})=0$
$\Leftrightarrow 2cos(2x-\frac{\pi}{6})cos(x+\frac{\pi}{6})=0$
$\Leftrightarrow cos (2x-\frac{\pi}{6})=0$ hoặc $cos(x+\frac{\pi}{6})=0$
$\Leftrightarrow 2x-\frac{\pi}{6}=k\pi+\frac{\pi}{2}$ hoặc $x+\frac{\pi}{6}=k\pi+\frac{\pi}{2}$
$\Leftrightarrow x=k\frac{\pi}{2}+\frac{\pi}{3}$ hoặc $x=k\pi+\frac{\pi}{3}$