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$(\sin x +3)(\sin^4 \frac{x}{2}- \sin^2 \frac{x}{2})+1=0$$\Leftrightarrow (\sin x +3) \sin^2 \frac{x}{2} (\sin^2 \frac{x}{2}-1) +1 =0$$\Leftrightarrow - (\sin x +3) \sin^2 \frac{x}{2}. \cos^2 \frac{x}{2} +1 =0$$\Leftrightarrow -(\sin x +3) \frac{(2 \sin \frac{x}{2}. \cos \frac{x}{2})^2}{4} +1 =0$$\Leftrightarrow -(\sin x +3) \frac{\sin^2 x}{4} +1 =0$$\Leftrightarrow -\sin^3 x -3 \sin^2 x +4 =0$

$(\sin x +3)(\sin^4 \frac{x}{2}- \sin^2 \frac{x}{2})+1=0$$\Leftrightarrow (\sin x +3) \sin^2 \frac{x}{2} (\sin^2 \frac{x}{2}-1) +1 =0$$\Leftrightarrow - (\sin x +3) \sin^2 \frac{x}{2}. \cos^2 \frac{x}{2} +1 =0$Do $\sin^2 a+ \cos^2 b =1 \Longrightarrow \sin^2 a -1 =\cos^2 b$$\Leftrightarrow -(\sin x +3) \frac{(2 \sin \frac{x}{2}. \cos \frac{x}{2})^2}{4} +1 =0$Do $\sin 2a = 2 \sin a. \cos a$$\Leftrightarrow -(\sin x +3) \frac{\sin^2 x}{4} +1 =0$$\Leftrightarrow -\sin^3 x -3 \sin^2 x +4 =0$

$(\sin x +3)(\sin^4 \frac{x}{2}- \sin^2 \frac{x}{2})+1=0$$\Leftrightarrow (\sin x +3) \sin^2 \frac{x}{2} (\sin^2 \frac{x}{2}-1) +1 =0$$\Leftrightarrow (\sin x +3) \sin^2 \frac{x}{2}. \cos^2 \frac{x}{2} +1 =0$$\Leftrightarrow (\sin x +3) \frac{(2 \sin \frac{x}{2}. \cos \frac{x}{2})^2}{4} +1 =0$$\Leftrightarrow (\sin x +3) \frac{\sin^2 x}{4} +1 =0$$\Leftrightarrow \sin^3 x +3 \sin^2 x +4 =0$

$(\sin x +3)(\sin^4 \frac{x}{2}- \sin^2 \frac{x}{2})+1=0$$\Leftrightarrow (\sin x +3) \sin^2 \frac{x}{2} (\sin^2 \frac{x}{2}-1) +1 =0$$\Leftrightarrow - (\sin x +3) \sin^2 \frac{x}{2}. \cos^2 \frac{x}{2} +1 =0$$\Leftrightarrow -(\sin x +3) \frac{(2 \sin \frac{x}{2}. \cos \frac{x}{2})^2}{4} +1 =0$$\Leftrightarrow -(\sin x +3) \frac{\sin^2 x}{4} +1 =0$$\Leftrightarrow -\sin^3 x -3 \sin^2 x +4 =0$