The equation ${\sin }^{4}x+{\cos }^{4}x+\sin 2x+\alpha =0$ is solvable for 

$\begin{array}{l}Wehave{\left({\sin }^{2}x+{\cos }^{2}x\right)}^2-2{\sin }^{2}x{\cos }^{2}x+\sin 2x+\alpha =0\\ 1-\frac{{\sin }^{2}2x}{2}+\sin 2x+\alpha =0\\ \Rightarrow {\sin }^{2}2x-2\sin 2x-2-2\alpha =0\\ \Rightarrow \sin 2x=\frac{2\pm \sqrt{4-4(-2-2\alpha )}}{2}\\ \Rightarrow \sin 2x=1\pm \sqrt{3+2\alpha }\\ \Rightarrow -1\le \sin 2x\le 1 \\ \Rightarrow -1\le 1\pm \sqrt{3+2\alpha }\text{\hspace{0.17em}≤1}\\ \Rightarrow -2\le \pm \sqrt{3+2\alpha }\le 0 \\ negecting positive sign\\ -2\le -\sqrt{3+2\alpha }\le 0\\ 4\ge 3+2\alpha \ge 0\\ \frac{-3}{2}\le \alpha \le \frac{1}{2}\end{array}$