# Number of maximal ideals in the ring $\mathbb{Z}_5[x]/\langle (x+1)^2(x+2)^3 \rangle$

Number of maximal ideals in the ring $$\mathbb{Z}_5[x]/\langle (x+1)^2(x+2)^3 \rangle$$ is

$$(a)$$ infinite

$$(b)3$$

$$(c)5$$

$$(d)2$$

I am aware of the correspondence theorem for rings. Following that idea, the maximal ideals in $$\mathbb{Z}_5[x]$$ containing $$\langle (x+1)^2(x+2)^3 \rangle$$ are $$\langle x+1 \rangle$$ and $$\langle x+2 \rangle$$ which should give two maximal ideals in the given quotient ring. Am I right? Will the field $$\mathbb{Z}_5$$ cause some trouble here?

Am I right? Will the field $$ℤ_5$$ cause some trouble here?
You have come to the correct conclusion. $$ℤ_5$$ helps you because then you are assured $$ℤ_5[x]$$ is a principal ideal domain.
Therefore the maximal ideals containing $$(x+1)^2(x+2)^3$$ correspond to the irreducible elements of its factorization, and you spotted those immediately.