\(\sqrt[3]{5+x}-x=5\)
\(\Leftrightarrow\sqrt[3]{5+x}=5+x\)
\(\Leftrightarrow5+x=125+75x+15x^2+x^3\)
\(\Leftrightarrow125+75x+15x^2+x^3-5-x=0\)
\(\Leftrightarrow120+74x+15x^2+x^3=0\)
\(\Leftrightarrow\left(x+4\right)\left(x+5\right)\left(x+6\right)=0\)
\(\Leftrightarrow\left[{}\begin{matrix}x=-4\\x=-5\\x=-6\end{matrix}\right.\)