\(x^4+x^3-4x^2+x+1=0\\ \Leftrightarrow\left(x^4-x^3\right)+\left(2x^3-2x^2\right)-\left(2x^2-2x\right)-\left(x-1\right)=0\\ \Leftrightarrow x^3\left(x-1\right)+2x^2\left(x-1\right)-2x\left(x-1\right)-\left(x-1\right)=0\\ \Leftrightarrow\left(x-1\right)\left(x^3+2x^2-2x-1\right)=0\\ \Leftrightarrow\left(x-1\right)\left[\left(x^3+3x^2+x\right)-\left(x^2+3x+1\right)\right]=0\\ \Leftrightarrow\left(x-1\right)\left[x\left(x^2+3x+1\right)-\left(x^2+3x+1\right)\right]=0\\ \Leftrightarrow\left(x-1\right)^2\left(x^2+3x+1\right)=0\)
\(\Leftrightarrow\left[{}\begin{matrix}\left(x-1\right)^2=0\\x^2+3x+1=0\end{matrix}\right.\\ \Leftrightarrow\Leftrightarrow\left[{}\begin{matrix}x-1=0\\4x^2+12x+4=0\end{matrix}\right.\\ \Leftrightarrow\left[{}\begin{matrix}x=1\\\left(4x^2+12x+9\right)-5=0\end{matrix}\right.\\ \Leftrightarrow\left[{}\begin{matrix}x=1\\\left(2x+3\right)^2-\sqrt{5^2}=0\end{matrix}\right.\\ \Leftrightarrow\left[{}\begin{matrix}x=1\\\left(2x+3-\sqrt{5}\right)\left(2x+3+\sqrt{5}\right)=0\end{matrix}\right.\)
\(\Leftrightarrow\left[{}\begin{matrix}x=1\\2x+3-\sqrt{5}=0\\2x+3+\sqrt{5}=0\end{matrix}\right.\\ \Leftrightarrow\left[{}\begin{matrix}x=1\\x=\dfrac{-3+\sqrt{5}}{2}\\x=\dfrac{-3-\sqrt{5}}{2}\end{matrix}\right.\)
\(\Leftrightarrow x^4-x^3+2x^3-2x^2-2x^2+2x-x+1=0\\ \Leftrightarrow\left(x-1\right)\left(x^3+2x^2-2x-1\right)=0\\ \Leftrightarrow\left(x-1\right)\left(x^3-x^2+3x^2-3x+x-1\right)=0\\ \Leftrightarrow\left(x-1\right)^2\left(x^2+3x+1\right)=0\\ \Leftrightarrow\left[{}\begin{matrix}x=1\\x=\dfrac{-3\pm\sqrt{5}}{2}\end{matrix}\right.\)
