\(x^4+x^3-27x-27=0\)
\(\Leftrightarrow x^3\left(x+1\right)-27\left(x+1\right)=0\)
\(\Leftrightarrow\left(x+1\right)\left(x^3-27\right)=0\)
\(\Leftrightarrow\left[{}\begin{matrix}x+1=0\\x^3=3^3\end{matrix}\right.\)
\(\Leftrightarrow\left[{}\begin{matrix}x=-1\\x=3\end{matrix}\right.\)