Help with understanding quotient ring structure

Question

Let $R$ be the ring $\mathbb{Z}[x]/((x^2+x+1)(x^3+x+1))$ and $I$ be the ideal generated by $2$ in $R$. What is the cardinality of the ring $R/I$?

I am having a hard time understanding what the ring $\mathbb{R}/I$ should be. I know the formal definitions of quotient ring and the ring operations in it. However I cant make much of the structure of quotient rings. Could anyone make this simple for me?

Answer 1

Hint: Instead of thinking about the quotient directly, first write it down in a familiar form using third isomorphism theorem. After that, use Chinese Remainder Theorem to conclude.

I'm travelling, so will be able to shed more light after a while.

Answer 2

This is the same structure as $ \mathbb{Z}_2[x]/((x^2 + x + 1)(x^3 + x + 1)) $. By the Chinese remainder theorem, we have

$$ \mathbb{Z}_2[x]/((x^2 + x + 1)(x^3 + x + 1)) \cong \mathbb{Z}_2[x]/(x^2 + x + 1) \times \mathbb{Z}_2[x]/(x^3 + x + 1) = \mathbb{F}_4 \times \mathbb{F}_8 $$

The cardinality of this structure is clearly $ 32 $.