\(\Leftrightarrow3\sqrt{x^2-\frac{1}{4}+\sqrt{\left(x+\frac{1}{2}\right)^2}}=\frac{1}{2}\left(2x+1\right)\left(x^2+1\right)\)
Do \(VT\ge0\Rightarrow VP\ge0\Rightarrow x\ge-\frac{1}{2}\)
\(\Leftrightarrow3\sqrt{x^2-\frac{1}{4}+x+\frac{1}{2}}=\frac{1}{2}\left(2x+1\right)\left(x^2+1\right)\)
\(\Leftrightarrow3\sqrt{\left(x+\frac{1}{2}\right)^2}=\frac{1}{2}\left(2x+1\right)\left(x^2+1\right)\)
\(\Leftrightarrow\frac{3}{2}\left(2x+1\right)=\frac{1}{2}\left(2x+1\right)\left(x^2+1\right)\)
\(\Leftrightarrow\left[{}\begin{matrix}2x+1=0\\x^2+1=3\end{matrix}\right.\) \(\Rightarrow\left[{}\begin{matrix}x=-\frac{1}{2}\\x=\sqrt{2}\end{matrix}\right.\)
