We prove that the titled equation has only finitely many solutions, generalizing a work by Li and Yuan and another work by Mao Hua Le. We apply our method to the specific Diophantine equation considered by Mao Hua Le and obtain all the solutions.
"On the Diophantine equation $h(a)x^2+f(a)=g(a)a^n$." Funct. Approx. Comment. Math. 61 (2) 201 - 206, December 2019. https://doi.org/10.7169/facm/1772