\[\begin{array}{l}
a)\,\,\,\,2{\sin ^2}x - \sin x\cos x - {\cos ^2}x = m\\
\Leftrightarrow 1 - \cos 2x - \frac{1}{2}\sin 2x - \frac{{1 + \cos 2x}}{2} = m\\
\Leftrightarrow \frac{1}{2} - \frac{3}{2}\cos 2x - \frac{1}{2}\sin 2x = m\\
\Leftrightarrow \sin 2x + 3\cos 2x = 1 - 2m\\
PT\,co\,nghiem\, \Leftrightarrow {1^2} + {3^2} \ge {\left( {1 - 2m} \right)^2}\\
\Leftrightarrow 4{m^2} - 4m - 9 \le 0 \Leftrightarrow \frac{{1 - \sqrt {10} }}{2} \le m \le \frac{{1 + \sqrt {10} }}{2}\\
b)\,\,Khi\,m = 1\,ta\,co:\\
\sin 2x + 3\cos 2x = - 1 \Leftrightarrow \sqrt {10} \left( {\frac{1}{{\sqrt {10} }}\sin 2x + \frac{3}{{\sqrt {10} }}\cos 2x} \right) = - 1\\
\Leftrightarrow \frac{1}{{\sqrt {10} }}\sin 2x + \frac{3}{{\sqrt {10} }}\cos 2x = - \frac{1}{{\sqrt {10} }}\\
Dat\,\left\{ \begin{array}{l}
\cos \alpha = \frac{1}{{\sqrt {10} }}\\
\sin \alpha = \frac{3}{{\sqrt {10} }}
\end{array} \right.\,ta\,duoc:\\
\sin 2x\cos \alpha + \cos 2x\sin \alpha = - \cos \alpha \\
\Leftrightarrow \sin \left( {2x + \alpha } \right) = \sin \left( {\alpha - \frac{\pi }{2}} \right)\\
\Leftrightarrow \left[ \begin{array}{l}
2x + \alpha = \alpha - \frac{\pi }{2} + k2\pi \\
2x + \alpha = \pi - \alpha + \frac{\pi }{2} + k2\pi
\end{array} \right.\\
\Leftrightarrow \left[ \begin{array}{l}
x = - \frac{\pi }{4} + k\pi \\
x = \frac{{3\pi }}{4} - \alpha + k\pi
\end{array} \right.
\end{array}\]
