I have a problem in which I have to calculate given two polynomials $f,g \in \mathbb{Q}[X]$ a polynomial with roots the product of the roots of $f$ and $g$. My reasoning shows that this reduces to calculate the resultant of the following two polynomials in $Y$: $$x^4+x^3y+x^2y^2+xy^3+y^4$$ and $$y^2+y+1$$ However, I don't want to compute a $6 \times 6$ determinant but I would rather prefer to work as the following property suggest ($R$ is the resultant):
Given $p,q \in A[X]$ with $r$ the euclidean remainder of $q$ by $p$, we have $R(p,q) = a_n^{m-gr(r)} R(p,r)$.
But my first polynomial is a polynomial in two variables. How can I deal with this situation?