\(f\left(x\right)=3x^2+x=0\)
\(\Leftrightarrow x\left(3x+1\right)=0\)
\(\Leftrightarrow\left[{}\begin{matrix}x=0\\3x+1=0\end{matrix}\right.\)
\(\Leftrightarrow\left[{}\begin{matrix}x=0\\3x=-1\end{matrix}\right.\)
\(\Leftrightarrow\left[{}\begin{matrix}x=0\\x=\dfrac{-1}{3}\end{matrix}\right.\)