Finding polynomials $g(x)$ such that $g(x)f(x)$ lies in an ideal

Question

For a given multivariate polynomial ring over a field, take an ideal $I$ and a polynomial $f(x)$, where $x$ can represent several variables. I am interested in finding all polynomials $g(x)$ such that $g(x)f(x) \in I$. Clearly, this is satisfied if $g(x) \in I$. If there exist functions $k(x)$ such that $k(x) f(x) = 0$ we should (I think) be able to add these functions as additional generators in $I$ to create more valid possibilities. However, it is not obvious to me whether these possibilities are exhaustive, or if there are others. In principle, it seems like one can come up with polynomials that are specifically tailored to the form of $f(x)$ such that $g(x) f(x) \in I$ but $g(x) a(x) \notin I$ for a generic $a(x)$, and I'm not sure if just finding the $k(x)$ described above is sufficient to account for all these cases.