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Decomposition of the product of irreducible elements

I’m stuck at the following problem:
Problem Let $f,g\in\mathbb{Z}[x]$ be irreducible polynomials, such that $f
e g$. Show that the product $fg$ can be expressed as a product of $f$ and $g$ with irreducible polynomials as factors.
Attempt: The decomposition of the product is simply given by the Chinese Remainder Theorem. Hence it is enough to show that the ring of polynomials over $\mathbb{Z}$ (or $\mathbb{Z}[x]$, doesn’t matter) is a UFD. It is not hard to show that the ring is an integral domain. So all we need to show is that it is a UFD.
Let $f,g\in\mathbb{Z}[x]$ be irreducible and let $A=\mathbb{Z}[x]/(f)$ and $B=\mathbb{Z}[x]/(g)$. I’m trying to show that the ideal $(f,g)$ is prime, but I don’t see why it should be so.
Let $\pi_A\colon \mathbb{Z}[x]\to A$ be the canonical homomorphism. We want to
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