\(\sqrt{2x+1}=2+\sqrt{x-3}\left(x\ge3\right)\)
\(\Rightarrow2x+1=4+x-3+4\sqrt{x-3}\Rightarrow x=4\sqrt{x-3}\)
\(\Rightarrow x^2=16\left(x-3\right)\Rightarrow x^2-16x+48=0\Rightarrow\left(x-12\right)\left(x-4\right)=0\)
\(\Rightarrow\left[{}\begin{matrix}x=12\\x=4\end{matrix}\right.\)