\(\left(x+1\right)^4=\left(x+1\right)^3\)
\(\Rightarrow\left(x+1\right)^4-\left(x+1\right)^3=0\)
\(\Rightarrow\left(x+1\right)^3\left(x+1-1\right)=0\)
\(\Rightarrow x\left(x+1\right)^3=0\)
\(\Rightarrow\left[{}\begin{matrix}x=0\\x=-1\end{matrix}\right.\)