\(4x^2+4x+24=4y^2\Leftrightarrow\left(2x+1\right)^2+23=\left(2y\right)^2\)
\(\Leftrightarrow\left(2x+1\right)^2-\left(2y\right)^2=-23\Leftrightarrow\left(2x+1-2y\right)\left(2x+1+2y\right)=-23\)
TH1: \(\left\{{}\begin{matrix}2x+1-2y=-23\\2x+1+2y=1\end{matrix}\right.\) \(\Rightarrow\left\{{}\begin{matrix}x=-6\\y=6\end{matrix}\right.\)
TH2: \(\left\{{}\begin{matrix}2x+1-2y=-1\\2x+1+2y=23\end{matrix}\right.\) \(\Rightarrow\left\{{}\begin{matrix}x=5\\y=6\end{matrix}\right.\)
TH3: \(\left\{{}\begin{matrix}2x+1-2y=1\\2x+1+2y=-23\end{matrix}\right.\) \(\Rightarrow\left\{{}\begin{matrix}x=-6\\y=-6\end{matrix}\right.\)
TH4: \(\left\{{}\begin{matrix}2x+1-2y=23\\2x+1+2y=-1\end{matrix}\right.\) \(\Rightarrow\left\{{}\begin{matrix}x=5\\y=-6\end{matrix}\right.\)