How to factorize a 4th degree polynomial? 
I need help to factorise the following polynomial:
$x^4 - 2x^3 + 8x^2 - 14x + 7$

The solution I need to reach is $(x-1)(x^3 - x^2 + 7x - 7)$. I need to factorize to this exactly as it is for a limit question where I cancel out the $(x-1)$ in the numerator and denominator. How do I proceed?

A brute force approach is to notice that
$$\begin{align} x^4-2x^3+8x^2-14x+7
=&(x^4-x^3)-(x^3-x^2)+7(x^2-2x+1)\\
=&x^3(x-1)-x^2(x-1)+7(x-1)^2\\
=&(x-1)(x^3-x^2+7x-7)\\
=&(x-1)\left(x^2(x-1)+7(x-1)\right)\\
=&(x-1)(x-1)(x^2+7)
\end{align}$$
QED
