How find this diophantine equation $(3x-1)^2+2=(2y^2-4y)^2+y(2y-1)^2-6y$ integer solution Find this following  Diophantine equation all integer solution
$$(3x-1)^2+2=(2y^2-4y)^2+y(2y-1)^2-6y$$
or
$$9x^2-6x+3=4y^4-12y^3+12y^2-5y$$
Maybe this equation can be solved by using Pell equation methods?
I want take right is Quadratic formula with $(ay^2+by+c)$
maybe we have
$$(3x-1)^2-A(ay^2+by+c)^2=B?$$


Yes, you can use the quadratic formula. Solve for $x$ and you get the form,
$$x = \frac{-b\pm\sqrt{f(y)}}{2a}$$
hence you must find integer $y$ such that,
$$f(y) = z^2$$
If your $f(y)$ is a polynomial only of degree $n=2$, then generally you can use a Pell equation to find an infinite more integer $y$. 
Unfortunately, your $f(y)$ has degree $n>2$ (namely $n=4$), and after some transformations this is an elliptic curve. And it is well-known an elliptic curve only has a finite number of integer solutions.
