$\begin{array}{l} - 1 \le \sin x \le 1\\ \Rightarrow - 2 \le 2\sin x \le 2\\ \Rightarrow 1 \le 2\sin x + 3 \le 5\\ \Rightarrow 1 \le \sqrt {2\sin x + 3} \le \sqrt 5 \\ \Rightarrow 2 \le 2\sqrt {2\sin x + 3} \le 2\sqrt 5 \\ \Rightarrow - 2 \le y \le 2\sqrt 5 - 4\\ \Rightarrow \left\{ \begin{array}{l} \max y = 2\sqrt 5 - 4 \Rightarrow \sin x = 1 \Rightarrow x = \frac{\pi }{2} + k2\pi \\ \min y = - 2 \Rightarrow \sin x = - 1 \Rightarrow x = - \frac{\pi }{2} + k2\pi \end{array} \right.\left( {k \in \mathbb{Z}} \right) \end{array}$
