\(4\sqrt{x+6}=x+1\)
\(\Leftrightarrow\left\{{}\begin{matrix}x+1\ge0\\\left(4\sqrt{x+6}\right)^2=\left(x+1\right)^2\end{matrix}\right.\)
\(\Leftrightarrow\left\{{}\begin{matrix}x\ge-1\\16\left(x^2+12x+36\right)=x^2+2x+1\end{matrix}\right.\)
\(\Leftrightarrow\left\{{}\begin{matrix}x\ge-1\\16x^2+192x+576=x^2+2x+1\end{matrix}\right.\)
\(\Leftrightarrow\left\{{}\begin{matrix}x\ge-1\\15x^2+190x+575=0\end{matrix}\right.\)
\(\Leftrightarrow\left\{{}\begin{matrix}x\ge-1\\\left[{}\begin{matrix}x=-5\\x=-\dfrac{23}{3}\end{matrix}\right.\end{matrix}\right.\)
\(\Leftrightarrow x\in\varnothing\)
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