$\begin{array}{l}
\cos 2x + 3\cos x - \sin x + 2 = 0\\
 \Leftrightarrow {\cos ^2}x - {\sin ^2}x + 3\cos x - \sin x + 2 = 0\\
 \Leftrightarrow {\cos ^2}x + 3\cos x - {\sin ^2}x - \sin x + 2 = 0\\
\Delta  = 9 + 4{\sin ^2}x + 4\sin x - 8 = 4{\sin ^2}x + 4\sin x + 1 = {\left( {2\sin x + 1} \right)^2}\\
 \Rightarrow \cos x = \dfrac{{ - 3 + 2\sin x + 1}}{2} = \sin x - 1,\\
\cos x = \dfrac{{ - 3 - 2\sin x - 1}}{2} =  - 2\sin x - 2\\
 \Leftrightarrow \left[ \begin{array}{l}
\cos x - \sin x =  - 1\\
\cos x + 2\sin x =  - 2
\end{array} \right. \Leftrightarrow \left[ \begin{array}{l}
\sqrt 2 \cos \left( {x + \dfrac{\pi }{4}} \right) =  - 1\\
\sqrt 5 \cos \left( {x - \alpha } \right) =  - 2
\end{array} \right.\left( {\alpha  = \arccos \dfrac{1}{{\sqrt 5 }}} \right)\\
 \Leftrightarrow \left[ \begin{array}{l}
x + \dfrac{\pi }{4} = \dfrac{{3\pi }}{4} + k2\pi \\
x + \dfrac{\pi }{{4x}} =  - \dfrac{{3\pi }}{4} + k2\pi \\
x - \alpha  = \arccos \left( {\dfrac{{ - 2}}{{\sqrt 5 }}} \right) + k2\pi \\
x - \alpha  =  - \arccos \left( { - \dfrac{2}{{\sqrt 5 }}} \right) + k2\pi 
\end{array} \right.\\
 \Leftrightarrow \left[ \begin{array}{l}
x = \dfrac{\pi }{2} + k2\pi \\
x =  - \pi  + k2\pi \\
x \approx 216,86 + k2\pi \\
x =  - \dfrac{\pi }{2} + k2\pi 
\end{array} \right. \Leftrightarrow \left[ \begin{array}{l}
x = \dfrac{\pi }{2} + k\pi \\
x =  - \pi  + k2\pi \\
x \approx 216,86 + k2\pi 
\end{array} \right.\left( {k \in Z} \right)\\
x \in \left( { - 49;2022} \right) \Rightarrow  - 49 < \dfrac{\pi }{2} + k\pi  < 2022\\
 \Leftrightarrow \dfrac{{ - 49}}{\pi } - \dfrac{1}{2} < k < \dfrac{{2022}}{\pi } - \dfrac{1}{2} \to 660gtx\\
x \in \left( { - 49;2022} \right) \Rightarrow  - 49 <  - \pi  + k2\pi  < 2022\\
 \Leftrightarrow \dfrac{{ - 49}}{\pi } <  - 1 + 2k < \dfrac{{2022}}{\pi }\\
 \Leftrightarrow \dfrac{{\dfrac{{ - 49}}{\pi } + 1}}{2} < k < \dfrac{{\dfrac{{2022}}{\pi } + 1}}{2} \to 330gtx\\
x \in \left( { - 49;2022} \right) \Rightarrow  - 49 < 216,86 + k2\pi  < 2022\\
 \Leftrightarrow \dfrac{{ - 49}}{{2\pi }} < 34,52 + k < \dfrac{{2022}}{{2\pi }} \Rightarrow  - 42,31 < k < 287,2\\
 \Rightarrow 330gtx\\
 \Rightarrow 1290gt
\end{array}$