Hướng dẫn giải : Chọn đáp án D

Đặt $t=\sin x+\cos x=\sqrt{2}\sin \left( x+\frac{\pi }{4} \right)(\left| t \right|\le \sqrt{2})$ ta có: $\sin x\cos x=\frac{{{t}^{2}}-1}{2}$

Khi đó $t+3.\frac{{{t}^{2}}-1}{2}-1=0\Leftrightarrow 3{{t}^{2}}+2t-5=0\Leftrightarrow \left[ \begin{align}& t=1 \\& t=\frac{-5}{3}\left( l \right) \\\end{align} \right.$

Với $t=1\Rightarrow \sin \left( x+\frac{\pi }{4} \right)=\sin \frac{\pi }{4}\Leftrightarrow \left[ \begin{align}& x+\frac{\pi }{4}=\frac{\pi }{4}+k2\pi  \\& x+\frac{\pi }{4}=\frac{3\pi }{4}+k2\pi  \\\end{align} \right.,k\in \mathbb{Z}\Leftrightarrow \left[ \begin{align}& x=k2\pi  \\& x=\frac{\pi }{2}+k2\pi  \\\end{align} \right.,k\in \mathbb{Z}$