<s>[INST] How to proceed with Euclidean algorithm for finding greatest common divisor of two polynomials. I am trying to find 
\begin{equation*}
gcd(x^4-x^3-4x^2-x+5,x^2+x-2). 
\end{equation*}
I have done the first step of long division and found.
\begin{equation*}
x^4-x^3-4x^2-x+5=(x^2-2x)(x^2+x-2)-5x+5
\end{equation*}
so we have 
\begin{equation*}
gcd(x^4-x^3-4x^2-x+5,x^2+x-2)=gcd(x^2+x-2,-5x+5)
\end{equation*}
now is where I am stuck. For the next step do we need to divide $x^2+x-2$ by $-5x+5$ or can we simply divide $x^2+x-2$ by $-x+1$ since the $gcd$ needs to a monomial?
Also if I did divide by $-x+1$ instead of $-5x+5$ would this change my procedure at all when reversing the algorithm to find the polynomials that multiply  $(x^4-x^3-4x^2-x+5,x^2+x-2)$ to give  $gcd(x^4-x^3-4x^2-x+5,x^2+x-2)$. (Bezout's lemma)
 [/INST] You are working with polynomials with coefficients in $\Bbb Q$ or in a larger field. $\Bbb Z[x]$ is not a principal ideal domain ($\langle x,2\rangle$ is not principal, for example), and the existence of $\gcd$ is not guaranteed.
That said, $5$ is an unit, and hence $-x+1$ and $-5x+5$ are associated. Therefore, you can change one by another when you want.
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