\[\begin{array}{l}
\frac{{{{\sin }^2}x + \sin x}}{{\sin x - 1}} = - 2\\
dk:\sin x \ne 1 \Leftrightarrow x \ne \frac{\pi }{2} + k2\pi (k \in Z)\\
\Leftrightarrow {\sin ^2}x + \sin x = - 2\sin x + 2\\
\Leftrightarrow {\sin ^2}x + 3\sin x - 2 = 0\\
= > \left[ \begin{array}{l}
\sin x = \frac{{ - 3 + \sqrt {17} }}{2}\\
\sin x = \frac{{ - 3 - \sqrt {17} }}{2}
\end{array} \right.\\
= > \left[ \begin{array}{l}
x = \arcsin \frac{{ - 3 + \sqrt {17} }}{2} + k2\pi \\
x = \pi - \arcsin \frac{{ - 3 + \sqrt {17} }}{2} + k2\pi \\
x = \arcsin \frac{{ - 3 - \sqrt {17} }}{2} + k2\pi \\
x = \pi - \arcsin \frac{{ - 3 - \sqrt {17} }}{2} + k2\pi
\end{array} \right.(k \in Z)(tm)
\end{array}\]

Vậy pt có nghiệm: \[S = \{ \arcsin \frac{{ - 3 + \sqrt {17} }}{2} + k2\pi ;\pi - \arcsin \frac{{ - 3 + \sqrt {17} }}{2} + k2\pi ;\arcsin \frac{{ - 3 - \sqrt {17} }}{2} + k2\pi ;\pi - \arcsin \frac{{ - 3 - \sqrt {17} }}{2} + k2\pi \} \]