\[\begin{array}{l}
\left( {m - 2} \right)\sin 2x = m + 1\,\,\,\left( * \right)\\
+ )\,\,TH1:\,\,\,m = 2\\
\Rightarrow \left( * \right) \Leftrightarrow 0\sin 2x = 3\\
\Rightarrow \left( * \right)\,\,\,VN.\\
+ )\,\,TH2:\,\,m \ne 2\\
\Rightarrow \left( * \right) \Leftrightarrow \sin 2x = \frac{{m + 1}}{{m - 2}}\\
\left( * \right)\,\,\,VN \Leftrightarrow \left[ \begin{array}{l}
\frac{{m + 1}}{{m - 2}} > 1\\
\frac{{m + 1}}{{m - 2}} < - 1
\end{array} \right. \Leftrightarrow \left[ \begin{array}{l}
\frac{{m + 1 - m + 2}}{{m - 2}} > 0\\
\frac{{m + 1 + m - 2}}{{m - 2}} < 0
\end{array} \right.\\
\Leftrightarrow \left[ \begin{array}{l}
\frac{3}{{m - 2}} > 0\\
\frac{{2m - 1}}{{m - 2}} < 0
\end{array} \right. \Leftrightarrow \left[ \begin{array}{l}
m > 2\\
\frac{1}{2} < m < 2
\end{array} \right..\\
KL.
\end{array}\]
