a) \(\sqrt{x+1}=x+1\)
\(\Leftrightarrow\left\{{}\begin{matrix}x+1\ge0\\x+1=\left(x+1\right)^2\end{matrix}\right.\) \(\Leftrightarrow\left\{{}\begin{matrix}x\ge-1\\x\left(x+1\right)=0\end{matrix}\right.\) \(\Leftrightarrow\left[{}\begin{matrix}x=0\\x=-1\end{matrix}\right.\)
b) \(\sqrt{3x^2+6x+3}=2x+1\)
\(\Leftrightarrow\sqrt{3\left(x^2+2x+1\right)}=2x+1\)
\(\Leftrightarrow\sqrt{3}\left|x+1\right|=2x+1\)
\(\Leftrightarrow\left[{}\begin{matrix}\sqrt{3}\left(x+1\right)=2x+1\\\sqrt{3}\left(x+1\right)=-2x-1\end{matrix}\right.\) \(\Leftrightarrow\left[{}\begin{matrix}\left(2-\sqrt{3}\right)x=\sqrt{3}-1\\\left(2+\sqrt{3}\right)x=-\sqrt{3}-1\end{matrix}\right.\)
\(\Leftrightarrow\left[{}\begin{matrix}x=\dfrac{\sqrt{3}-1}{2-\sqrt{3}}\\x=\dfrac{-\sqrt{3}-1}{2+\sqrt{3}}\end{matrix}\right.\)
c) \(\sqrt{2x^2+x+3}=-x-5\)
\(\Leftrightarrow\left\{{}\begin{matrix}-x-5\ge0\\2x^2+x+3=\left(-x-5\right)^2\end{matrix}\right.\)
\(\Leftrightarrow\left\{{}\begin{matrix}x\le-5\\2x^2+x+3=x^2+10x+25\end{matrix}\right.\)
\(\Leftrightarrow\left\{{}\begin{matrix}x\le-5\\x^2-9x-22=0\end{matrix}\right.\)
\(\Leftrightarrow\left\{{}\begin{matrix}x\le-5\\x=-2\cup x=11\end{matrix}\right.\) \(\Leftrightarrow x=-2\)
d) \(\sqrt{x^2-4x-5}=\sqrt{2x^2+3x+1}\)
\(\Leftrightarrow\left\{{}\begin{matrix}x^2-4x-5\ge0\\x^2-4x-5=2x^2+3x+1\end{matrix}\right.\)
\(\Leftrightarrow\left\{{}\begin{matrix}x\le-1\cup x\ge5\\x^2-7x+6=0\end{matrix}\right.\)
\(\Leftrightarrow\left\{{}\begin{matrix}x\le-1\cup x\ge5\\x=1\cup x=6\end{matrix}\right.\) \(\Leftrightarrow x=6\)
