Consider the quotient ring $\mathbb{Z}[x]/(x^2-3, 2x+4)$.  Sometimes we can get identifications of rings like these with other more familiar rings if we first kill $x^2-3$ in the ring $\mathbb{Z}[x]$ and then kill $2x+4$ or vise versa.  The substitution homomorphism $\mathbb{Z}[x] \rightarrow \mathbb{Z}[-2]$ doesn't seem to be relevant since its ideal is $x + 2$ and not $2x+4$.  Working with $x^2 -3$ though I got that $\mathbb{Z}[x]/(x^2-3)\cong \mathbb{Z}[\sqrt{3}]$ and since the substitution homomorphism is surjective, we have that $\mathbb Z[x](x^2-3, 2x+4) \cong \mathbb{Z}[\sqrt{3}]/(2\sqrt{3} + 4)$, making use of the correspondence theorem.  But my use isn't good enough since I thought the identification would be a more familiar ring.
This problem is from a practice exam I was working on.     What is the cardinality of the quotient $\mathbb{Z}[x]/(x^2-3,2x+4)$ ?   Thoughts. If I find a ring that is easier to handle then this then I can go from there. So I think this is isomorphic to $\mathbb{Z}_2[x]/(x^2-3,x+2)$. And then I am stuck.  Am I correct so far? What should I be looking to do from here?  Thanks for your time and your answers.
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