\(x^3-12x-16=0\Leftrightarrow x^2\left(x+2\right)-2x\left(x+2\right)-8\left(x+2\right)=0\)
\(\Leftrightarrow\left(x+2\right)\left(x^2-2x-8\right)=0\)
\(\Leftrightarrow\left(x+2\right)\left[x\left(x-4\right)+2\left(x-4\right)\right]=0\)
\(\Leftrightarrow\left(x+2\right)^2\left(x-4\right)=0\Leftrightarrow\orbr{\begin{cases}x=-2\\x=4\end{cases}}\)