The number of solution of ${\sin }^4\mathrm{x}-{\cos }^2\mathrm{xsin}\mathrm{x}+2{\sin }^2\mathrm{x}+\sin \mathrm{x}=0\text{ in }0\le \mathrm{x}\le 3\mathrm{\pi }$ is

$\begin{array}{l}{\sin }^4\mathrm{x}-{\cos }^2\mathrm{xsin}\mathrm{x}+2{\sin }^2\mathrm{x}+\sin \mathrm{x}=0\\ \text{ or }\sin \mathrm{x}\left[{\sin }^3\mathrm{x}-{\cos }^2\mathrm{x}+2\sin \mathrm{x}+1\right]=0\\ \text{ or }\sin \mathrm{x}\left[{\sin }^3\mathrm{x}-1+{\sin }^2\mathrm{x}+2\sin \mathrm{x}+1\right]=0\\ \text{ or }\sin \mathrm{x}\left[{\sin }^3\mathrm{x}+{\sin }^2\mathrm{x}+2\sin \mathrm{x}\right]=0\\ \text{ or }{\sin }^2\mathrm{x}=0\text{ or }{\sin }^2\mathrm{x}+\sin \mathrm{x}+2=0\end{array}$

(not possible for real x)
or  sinx =0

Hence, the solutions are $\mathrm{x}=0,\mathrm{\pi },2\mathrm{\pi },3\mathrm{\pi }$