Number of solutions of $3\sin^2 x+\cos^2 x+\sqrt{3} \sin x+\cos x+1=\sqrt{3}\sin x\cos x$ Find Number of solutions of 

$$3\sin^2 x+\cos^2 x+\sqrt{3} \sin x+\cos x+1=\sqrt{3}\sin x\cos x$$ in $\left[0 \:\: 10\pi\right]$

My Try:
The given equation is
$$2+2\sin^2 x+2\sin\left(x+\frac{\pi}{6}\right)=\frac{\sqrt{3}}{2}\sin (2x)$$ $\implies$
$$3+2\sin\left(x+\frac{\pi}{6}\right)=\frac{\sqrt{3}}{2}\sin (2x)+\cos (2x)$$
Any idea here?

We can rewrite the equation as $\left(\sqrt 3 \sin x - \cos x \right)^2+ \left(1+ \sqrt 3 \sin x \right) \left(1+\cos x\right) = 0$
Let $a=1+ \sqrt 3 \sin x$ and $b=1+\cos x$
Then the equation is $(a-b)^2+ab = 0 \Rightarrow a^2-ab+b^2 =0$
But $a^2 -ab+b^2 = 0 \iff a=0,b=0 $ which is not possible.
Hence the equation has no solutions.
