\(\Leftrightarrow x^2+x+2-1=\left(2-x\right)\left(\sqrt{x^2+x+2}-1\right)\)
\(\Leftrightarrow\left(\sqrt{x^2+x+2}-1\right)\left(\sqrt{x^2+x+2}+1\right)-\left(2-x\right)\left(\sqrt{x^2+x+2}-1\right)=0\)
\(\Leftrightarrow\left(\sqrt{x^2+x+2}-1\right)\left(\sqrt{x^2+x+2}+1-2+x\right)=0\)
\(\Leftrightarrow\left[{}\begin{matrix}\sqrt{x^2+x+2}=1\left(1\right)\\\sqrt{x^2+x+2}=1-x\left(2\right)\end{matrix}\right.\)
\(\left(1\right)\Leftrightarrow x^2+x+1=0\left(vn\right)\)
\(\left(2\right)\Leftrightarrow\left\{{}\begin{matrix}1-x\ge0\\x^2+x+2=\left(1-x\right)^2\end{matrix}\right.\)
\(\Leftrightarrow\left\{{}\begin{matrix}x\le1\\3x=-1\end{matrix}\right.\) \(\Rightarrow x=-\frac{1}{3}\)
