A ternary quadratic non-homogeneous diophantine equation in $\mathbb Z[t]$

Question

I am interested in the diophantine equation in $\mathbb Z[t]$:

$$6Z^2 + 5((t + 1)X + tY − 1)Z +((t + 1)X + tY − 1)^2+ XY = 0$$ (the unknown variables are $X,Y,Z$)

Can one determine ALL the solution in $\mathbb Z[t]$?

Thanks in advance

Answer 1

The solution to the equation.

$$6Z^2+5((t+1)X+tY-1)Z+((t+1)X+tY-1)^2+XY=0$$

Can be written as an ordinary quadratic equation.

$$Z=\frac{-5((t+1)X+tY-1)\pm{D}}{12}$$

Where.

$$D^2=25((t+1)X+tY-1)^2-24((t+1)X+tY-1)^2-24XY$$

$$((t+1)X+tY-1+D)((t+1)X+tY-1-D)=24XY$$

The solution of this equation there. https://mathoverflow.net/questions/215197/quadratic-diophantine-equation-in-mathbb-zt