\(\Leftrightarrow\left(x-2\right)\left(2x+5\right)-\left(x-2\right)\sqrt{x^2+2x+2}=0\)
\(\Leftrightarrow\left(x-2\right)\left(2x+5-\sqrt{x^2+2x+2}\right)=0\)
\(\Leftrightarrow\left[{}\begin{matrix}x=2\\2x+5=\sqrt{x^2+2x+2}\left(1\right)\end{matrix}\right.\)
\(\left(1\right)\Leftrightarrow\left\{{}\begin{matrix}2x+5\ge0\\x^2+2x+2=\left(2x+5\right)^2\end{matrix}\right.\)
\(\Leftrightarrow\left\{{}\begin{matrix}x\ge-\dfrac{5}{2}\\3x^2+18x+23=0\end{matrix}\right.\)
\(\Rightarrow\left[{}\begin{matrix}x=\dfrac{-9+2\sqrt{3}}{3}\\x=\dfrac{-9-2\sqrt{3}}{3}< -\dfrac{5}{2}\left(loại\right)\end{matrix}\right.\)
\(\Rightarrow\left\{{}\begin{matrix}a=-9\\b=2\end{matrix}\right.\) \(\Rightarrow a^2+b^2=85\)
