<s>[INST] Find the greatest common divisor of $f(x)=2x^3+2x^2+x+4$ and $g(x)=x^4+3x^3+4x^2+3x$ Find the greatest common divisor of $f(x)=2x^3+2x^2+x+4$ and $g(x)=x^4+3x^3+4x^2+3x$ in $\mathbb{Z}_{11}[x]$

 [/INST] We have to perform the Euclidean algorithm modulo $11$:
$$\begin{align}
h_1(x)&:=g(x)+(5x-1)f(x)\equiv 7x^2+7,\\
h_2(x)&:=f(x)+(6x+6)h_1(x)\equiv -x+2,\\
h_3(x)&:=h_1(x)+(7x+3)h_2(x)\equiv 2.\\
\end{align}$$ Hence, in $\mathbb{Z}_{11}[x]$,
$$\gcd(g(x),f(x))=\gcd(h_1(x),f(x))=\gcd(h_1(x),h_2(x))=\gcd(h_3(x),h_2(x))=1.$$
</s>