\(x+2\sqrt{2}x^2+2x^3=0\)
\(\Leftrightarrow x\left(1+2\sqrt{2}x+2x^2\right)=0\)
\(\Leftrightarrow x\left(1+\sqrt{2}x\right)^2=0\)
\(\Leftrightarrow\orbr{\begin{cases}x=0\\\left(1+\sqrt{2}x\right)^2=0\end{cases}}\Leftrightarrow\orbr{\begin{cases}x=0\\1+\sqrt{2}x=0\end{cases}}\Leftrightarrow\orbr{\begin{cases}x=0\\x=-\frac{1}{\sqrt{2}}\end{cases}}\)
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