Finding a generator for the product of two ideals in $ \mathbb{C}[X][Y]$

Question

Let $f, g\in\mathbb{C}[X,Y]$ be two irreducible elements in $\mathbb{C}[X,Y]$ such that $(f) \neq (g)$. Give a generator for the ideal $((f) + (g)) \cdot ((f) \cap (g))$.

I figured out that $(f) \cap (g) = (fg)$, because $\mathbb{C}[X,Y]$ is a UFD and so $(f) \cap (g) = (\operatorname{lcm}(f, g))$, but I haven’t been able to figure out the rest.

Any help is greatly appreciated.