Doubt related to finding the gcd between two elements of an euclidian domain.

This is a pretty elementary doubt, I assume, but anyway:

The doubt. Let's say we want to find $gcd(a,b)$, where $a,b \in D$, where $D$ represents an eucilidain domain. Can we use rests and quotients that are NOT in $D$? I leave a pratical example below.

Pratical Example. Let's say we are working on $\mathbb{Z}_2[x]$. Determine $gcd(x^3+x,x^2+x)$.

What I've done. As $dg(x^3+x) > dg(x^2+x)$ (where $dg$ stands for degree) we proceed the following way:

And so the answer would be $\widetilde{\frac{1}{2}(x+1)} = \widetilde{x+1} \in Z_2[x]$. My question is basically why is this computation not right. My guess is that I messed up by not using some elements NOT in $Z_2[x]$ during the process but I would like more details on it. Thanks for all the help in advance.

In $\Bbb Z_2[x]$ you have $2x=0x=0$, so the first division leaves no remainder.