\(x^4-2x^3-x^2-2x+1=0\)
\(\Leftrightarrow x^4-3x^3+x^2+x^3-3x^2+x+x^2-3x+1=0\)
\(\Leftrightarrow x^2\left(x^2-3x+1\right)+x\left(x^2-3x+1\right)+\left(x^2-3x+1\right)=0\)
\(\Leftrightarrow\left(x^2-3x+1\right)\left(x^2+x+1\right)=0\)
\(\Leftrightarrow\orbr{\begin{cases}x^2-3x+1=0\\x^2+x+1=0\end{cases}}\)\(\Leftrightarrow\orbr{\begin{cases}\left(x-\frac{3}{2}\right)^2-\frac{5}{4}=0\\\left(x+\frac{1}{2}\right)^2+\frac{3}{4}>0\forall x\end{cases}}\)
\(\Leftrightarrow\left(x-\frac{3}{2}\right)^2-\frac{5}{4}=0\)\(\Leftrightarrow\left(x-\frac{3}{2}\right)^2=\frac{5}{4}\)
\(\Leftrightarrow x-\frac{3}{2}=\pm\sqrt{\frac{5}{4}}\)\(\Leftrightarrow x=\pm\frac{\sqrt{5}}{2}+\frac{3}{2}\)
