An elementary algorithm for solving a diophantine equation of degree four with Runge's condition

We propose an elementary algorithm for solving a diophantine equation
\begin{equation*} (p(x,y)+a_1x+b_1y)(p(x,y)+a_2x+b_2y)-dp(x,y)-a_3x-b_3y-c=0 \tag{*} \end{equation*}
of degree four, where $p(x,y)$ denotes an irreducible quadratic form of positive discriminant and $(a_1,b_1) \neq (a_2,b_2)$. The last condition guarantees that the equation $(*)$ can be solved using the well known Runge's method, but we prefer to avoid the use of any power series that leads to upper bounds for solutions useless for a computer implementation.