Đặt \(x+2=a\)
\(\Rightarrow\sqrt[3]{a+1}+\sqrt[3]{a}+\sqrt[3]{a-1}=0\)
\(\Leftrightarrow\sqrt[3]{a+1}+\sqrt[3]{a-1}=-\sqrt[3]{a}\)
\(\Leftrightarrow2a+3\sqrt[3]{a^2-1}.\left(-\sqrt[3]{a}\right)=a\)
\(\Leftrightarrow3\sqrt[3]{a^3-a}=a\)
\(\Leftrightarrow27\left(a^3-a\right)=a^3\)
\(\Leftrightarrow a\left(26a^2-27\right)=0\)
\(\Rightarrow\left[{}\begin{matrix}a=0\\a=-\sqrt{\frac{27}{26}}\\a=\sqrt{\frac{27}{26}}\end{matrix}\right.\) \(\Rightarrow\left[{}\begin{matrix}x=-2\\x=-\sqrt{\frac{27}{26}}-2\\x=\sqrt{\frac{27}{26}}-2\end{matrix}\right.\)
