\(x^2+x+6=y^2\)
\(\Leftrightarrow x^2+x+6-y^2=0\)
\(\Leftrightarrow4x^2+4x+24-4y^2=0\)
\(\Leftrightarrow\left(4x^2+2x+4xy\right)+\left(2x+1+2y\right)-\left(4xy+2y+4y^2\right)+23=0\)
\(\Leftrightarrow2x\left(2x+1+2y\right)+\left(2x+1+2y\right)-2y\left(2x+1+2y\right)=-23\)
\(\Leftrightarrow\left(2x+1+2y\right)\left(2x+1-2y\right)=-23\)
\(\Leftrightarrow\left[{}\begin{matrix}\left\{{}\begin{matrix}2x+1+2y=1\\2x+1-2y=-23\end{matrix}\right.\\\left\{{}\begin{matrix}2x+1+2y=-1\\2x+1-2y=23\end{matrix}\right.\\\left\{{}\begin{matrix}2x+1+2y=-23\\2x+1-2y=1\end{matrix}\right.\\\left\{{}\begin{matrix}2x+1+2y=23\\2x+1-2y=-1\end{matrix}\right.\end{matrix}\right.\)
\(\Leftrightarrow\left[{}\begin{matrix}\left\{{}\begin{matrix}x=-6\\y=6\end{matrix}\right.\\\left\{{}\begin{matrix}x=5\\y=-6\end{matrix}\right.\\\left\{{}\begin{matrix}x=-6\\y=-6\end{matrix}\right.\\\left\{{}\begin{matrix}x=5\\y=6\end{matrix}\right.\end{matrix}\right.\)
