The range of $\alpha$ for which the equation 

${\sin }^4 \mathrm{x}+{\cos }^4 \mathrm{x}+\sin 2\mathrm{x}+\mathrm{\alpha }=0$ has solution

$$\begin{gathered} {\sin }^4\mathrm{x}+{\cos }^4\mathrm{x}+\sin 2\mathrm{x}+\mathrm{\alpha }=0 \\ 1-2{\sin }^2\mathrm{x} {\cos }^2\mathrm{x}+\sin 2\mathrm{x}+\mathrm{\alpha }=0 \\ 1-\frac{1}{2}{\sin }^{2}2\mathrm{x}+\sin 2\mathrm{x}+\mathrm{\alpha }=0 \\ \mathrm{\alpha }=\frac{1}{2}{\sin }^{2}2\mathrm{\alpha }-\sin 2\mathrm{x}-1 \\ =\frac{1}{2}{(\sin 2\mathrm{x}-1)}^2-\frac{3}{2} =\frac{1}{2}4-\frac{3}{2}=\frac{1}{2} \end{gathered}$$

Min. value at $\sin 2\mathrm{x}=1$

$=\frac{-3}{2}$

Range of  $\mathrm{\alpha }=\left[\raisebox{1ex}{$-3$}\!\left/ \!\raisebox{-1ex}{$2$}\right., \frac{1}{2}\right]$