On the variant \(Q\left(n!\right)=P\left(x\right)\) of the Brocard–Ramanujan Diophantine equation

Let \({P}_{1},{P}_{2},\ldots,{P}_{l}\in {\mathbb{Z}}\left[X,Y\right]\) $P_1,P_2,\dots ,P_l\in \mathbb{Z}\left[X,,,Y\right]$ be homogeneous polynomials of degree 2, with nonpositive discriminant and let Q be a polynomial with integer coefficients. Under the assumption that 0 is a root of Q with odd multiplicity, we show that the variant of the Brocard–Ramanujan Diophantine equation \(Q\left(n!\right)=\prod_{i=1}^{l}{P}_{i}{\left({X}_{i},{Y}_{i}\right)}^{{\alpha }_{i}}\) $Q\left(n,!\right)={\prod }_{i=1}^l{P}_i{\left(X_i,,,Y_i\right)}^{{\alpha }_i}$ has only finitely many integer solutions \(\left(n,{\overline{X}}_{l},{\overline{Y}}_{l},{\overline{\alpha }}_{l}\right),\) $\left(n,,,{\overline{X}}_l,,,{\overline{Y}}_l,,,{\overline{\alpha }}_l\right),$ where \({\overline{X}}_{l}=\left({X}_{1},{X}_{2},\ldots {X}_{l}\right)\) ${\overline{X}}_l=\left(X_1,,,X_2,,,\dots ,X_l\right)$ .