On a generalization of the Brocard--Ramanujan Diophantine equation

Let $Q_1,\mathrm{.}\mathrm{.}\mathrm{.},Q_r\in \mathbb{Z}[x]$ be polynomials having $0$ as a root. Let $f(x,y)\in \mathbb{Z}[x,y]$ be a homogeneous polynomial with factorization $f(x,y)=f_1(x,y{)}^{e_1}\cdots f_u(x,y{)}^{e_u}$, where $f_i(x,y)$ are irreducible homogeneous polynomials of degree $d_i\ge 2$. Fix some positive integers $A_1,\mathrm{.}\mathrm{.}\mathrm{.},A_r$. We show that under certain conditions, the diophantine equation ${\prod }_{i=1}^r{Q}_i(A_i^{n_i}{n}_i!)=f(x,y)$ has finitely many integer solutions.