complex analysis - How to calculate the polynomial coefficients of a fraction of polynomials?

I have a polynomial fraction which results in a polynomial

$$\frac{f(x)}{g(x)}=q(x)$$
with $f$ $g$ and $q$ being polynomials. I have formulas for the coefficients of $f(x)$ and $g(x)$ dependent on the degree of $f$ and of $g$.

Now I searched for a way to express the coefficients of $q(x)$ by algebraic expressions of the coefficients of $f$ and $g$.

One way I think I found until now is the "subresultant PRS algorithm" which allows to calculate the coefficients of $q(x)$ by appropriate determinants of matrices with coefficients of $f$ and $g$.

But these determinants seem not to be calculable in a non-computeralgebra situation.

Are there other methods ( e.g. algebraic calculus complex analysis ) how to

tackle such a general problem ?