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?

Notice that the coefficients add to $0$, hence $p(1) = 0$. (Instead of noticing this possibly hard to see fact, we might have also chosen to try $1$ because of the rational root theorem)
Now we can use polynomial long division using $x-1$. I'm partial to this method that is essentially the same thing but presented as factoring by grouping.
$x^3(x-1) - x^2 (x-1) + 7x(x - 1) - 7(x - 1) = (x-1)(x^3 - x^2 + 7x - 7)$.
You say this is the solution you need to reach, but we can go further:
We see that the roots still add to zero, so  we can do the same process:
$(x-1)(x^2(x-1) + 7(x-1)) = (x-1)^2 (x^2 + 7)$
