\(\left(x+1\right)^{2y}=1024\)
\(\Rightarrow\left[\left(x+1\right)^y\right]^2=32^2\)
\(\Rightarrow\left(x+1\right)^y=32\)
Do \(x,y\in Z\)
\(\Rightarrow\left\{{}\begin{matrix}x+1=2\\y=5\end{matrix}\right.\)\(\Rightarrow\left\{{}\begin{matrix}x=1\\y=5\end{matrix}\right.\)