$\begin{array}{l} \sin x\left( {2{{\cos }^3}x + \sin x + \cos x} \right) = 1 + 2{\sin ^2}x{\cos ^2}x\\ \Leftrightarrow 2\sin x{\cos ^3}x + {\sin ^2}x + \sin x\cos x = 1 + 2{\sin ^2}x{\cos ^2}x\\ \Leftrightarrow 2\sin x{\cos ^3}x - 2{\sin ^2}x{\cos ^2}x + {\sin ^2}x + \sin x\cos x - 1 = 0\\ \Leftrightarrow 2\sin x{\cos ^2}x\left( {\cos x - \sin x} \right) - {\cos ^2}x + \sin x\cos x = 0\\ \Leftrightarrow 2\sin x{\cos ^2}x\left( {\cos x - \sin x} \right) + \cos x\left( {\sin x - \cos x} \right) = 0\\ \Leftrightarrow \left( {\sin x - \cos x} \right)\left( {\cos x - 2\sin x{{\cos }^2}x} \right) = 0\\ \Leftrightarrow \cos x\left( {\sin x - \cos x} \right)\left( {1 - 2\sin x\cos x} \right) = 0\\ \Leftrightarrow \cos x\left( {\sin x - \cos x} \right)\left( {{{\sin }^2}x + {{\cos }^2}x - 2\sin x\cos x} \right) = 0\\ \Leftrightarrow \cos x\left( {\sin x - \cos x} \right){\left( {\sin x - \cos x} \right)^2} = 0\\ \Leftrightarrow \cos x{\left( {\sin x - \cos x} \right)^3} = 0\\ \Leftrightarrow \left[ \begin{array}{l} \cos x = 0\\ \sin x - \cos x = 0 \end{array} \right.\\ \Leftrightarrow \left[ \begin{array}{l} \cos x = 0\\ \sqrt 2 \sin \left( {x - \dfrac{\pi }{4}} \right) = 0 \end{array} \right. \Leftrightarrow \left[ \begin{array}{l} x = \dfrac{\pi }{2} + k\pi \\ x - \dfrac{\pi }{4} = k\pi \end{array} \right.\\ \Leftrightarrow \left[ \begin{array}{l} x = \dfrac{\pi }{2} + k\pi \\ x = \dfrac{\pi }{4} + k\pi \end{array} \right.\left( {k \in \mathbb{Z}} \right) \end{array}$
