<s>[INST] 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?
 [/INST] Let $c = \cos(x)$ and $s = \sin(x)$, solve for $c$:
$$3s^2 + c^2 + \sqrt 3 s + c + 1 = \sqrt 3 cs$$
$$c^2 + (1 - \sqrt 3 s)c + 3s^2 + \sqrt 3 s + 1 = 0$$
$$c = \frac{\sqrt 3 s - 1 \pm \sqrt{ (1 - \sqrt 3 s)^2 - 4(3s^2 + \sqrt 3 s + 1)}}{2}$$
Normally here you'd check the range on the discriminate, but the discriminate is a perfect square, so:
$$c = \frac{\sqrt 3 s - 1 \pm \sqrt{ -9s^2 - 6\sqrt 3 s - 3}}{2}$$
$$c = \frac{\sqrt 3 s - 1 \pm i\sqrt{(3s + \sqrt 3)^2}}{2}$$
$$c = \frac{\sqrt 3 s - 1 \pm i|3s + \sqrt 3|}{2}$$
So if $s$ is real then $c$ is complex, so no solutions.
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