$\begin{array}{l} y = \left( {2\sin x + \cos x} \right)\left( {3\sin x - \cos x} \right)\\ y = 6{\sin ^2}x - 2\sin x\cos x + 3\sin x\cos x - {\cos ^2}x\\ y = 7{\sin ^2}x + \sin x\cos x - \left( {{{\sin }^2}x + {{\cos }^2}x} \right)\\ y = 7{\sin ^2}x + \sin x\cos x - 1\\ y = \dfrac{{7\left( {1 - \cos 2x} \right)}}{2} + \dfrac{1}{2}\sin 2x - 1\\ y = \dfrac{{ - 7}}{2}\cos 2x + \dfrac{1}{2}\sin 2x + \dfrac{5}{2}\\ y = \dfrac{1}{2}\left( {\sin 2x - 7\cos 2x} \right) + \dfrac{5}{2}\\ y = \dfrac{1}{2}.\sqrt {50} \left( {\dfrac{1}{{\sqrt {50} }}\sin 2x - \dfrac{7}{{\sqrt {50} }}\cos 2x} \right) + \dfrac{5}{2}\\ y = \dfrac{{\sqrt {50} }}{2}\sin \left( {x - \alpha } \right) + 3\left( {\alpha = \arccos \dfrac{1}{{\sqrt {50} }}} \right)\\ - 1 \le \sin \left( {x - \alpha } \right) \le 1\\ \Rightarrow \dfrac{5}{2} - \dfrac{{\sqrt {50} }}{2} \le y \le \dfrac{5}{2} + \dfrac{{\sqrt {50} }}{2}\\ \Rightarrow \left\{ \begin{array}{l} \max y = \dfrac{5}{2} + \dfrac{{\sqrt {50} }}{2}\\ \min y = \dfrac{5}{2} - \dfrac{{\sqrt {50} }}{2} \end{array} \right. \end{array}$
