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\(\Leftrightarrow2\sqrt{2x+4+2\sqrt{2x+3}}=2\left(x+1\right)\)
\(\Leftrightarrow\sqrt{2x+3+2\sqrt{2x+3}+1}=x+1\)
\(\Leftrightarrow\sqrt{\left(\sqrt{2x+3}+1\right)^2}=x+1\)
\(\Leftrightarrow\sqrt{2x+3}+1=x+1\)
\(\Leftrightarrow\sqrt{2x+3}=x\left(x\ge0\right)\)
\(\Leftrightarrow2x+3=x^2\)
\(\Leftrightarrow x^2-2x-3=0\Rightarrow\left[{}\begin{matrix}x=-1\left(l\right)\\x=3\end{matrix}\right.\)
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