<s>[INST] Identify $\mathbb{Z}[x]/(x^2-3,2x+4)$. I need help to identify $\mathbb{Z}[x]/(x^2-3,2x+4)$.
I've been solving such problems in an approach like:
$$
2(x^2-3)=2x^2-6, x(2x+4)=2x^2+4x \\
(2x^2+4x)-(2x^2-6)=4x+6, 2(2x+4)=4x+8 \\
(4x+8)-(4x+6)=2
$$
What shall I do next, please? Thank you.
Simon
 [/INST] we have $2=2(x^2-3)+(2-x)(2x+4)$ and then $(x^2-3,2x+4,2)=(2,x^2-3)$
$\mathbb Z[x]/(x^2-3,2x+4) \cong\mathbb Z[x]/(2,x^2-3)\cong \mathbb Z_2[x]/(x^2-1) \cong  \mathbb Z_2[x]/(x-1)^2 \cong \mathbb Z_2[x]/(x)^2 $
</s>