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