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?

Expand out the difference (left side - right side) as 
$$ F(x) = 4-2\,  \cos^2 \left( x \right)  +\sqrt {3}\sin \left( 
x \right) +\cos \left( x \right) -\sqrt {3}\sin \left( x \right) \cos
 \left( x \right) 
$$
If $\sin(x) = s$, $\cos(x) = \pm \sqrt{1-s^2}$, and
$$ F(x) = 2 + 2 s^2 + \sqrt{3} s \pm (1-\sqrt{3} s) \sqrt{1-s^2}$$
In order for that to be $0$, we'd need
$$ \sqrt{1-s^2} =  \mp \frac{2 s^2 + \sqrt{3} s + 2}{1-\sqrt{3} s} $$
and thus $$1 - s^2 = \dfrac{(2s^2 + \sqrt{3} s + 2)^2}{(1-\sqrt{3} s)^2}$$
so that
$$ 7\,{s}^{4}+2\,\sqrt {3}{s}^{3}+6\,\sqrt {3}s+9\,{s}^{2}+3 = 0$$
This quartic has positive discriminant ($209088$), so its roots are either all real or all non-real.  In this case they turn out to be all non-real.
