Hi, think this is the correct forum, apologies if not.

Find the highest common factor of the polynomials

$\displaystyle x^5 - 2x^4 - 3x^3 + 13x^2 -16x + 6$ and $\displaystyle x^4 - 2x^3 - 2x^2 + 8x - 8$.

Hence factorise $\displaystyle x^4 - 2x^3 - 2x^2 + 8x - 8$ into irreducible polynomials in $\displaystyle R[x]$.

First I started off by factorising, by long division I got the following:

$\displaystyle x^5 - 2x^4 - 3x^3 + 13x^2 -16x + 6 = x(x^4 - 2x^3 - 2x^2 + 8x - 8) + (-x^3 +5x^2-8x+6)$

Not sure where to go from here. I'm ok with the Euclidean algorithm but not sure how to apply it here?

Thanks in advance