Solving a diophantine equation (2nd power) with Pell's equation

Hello, people,

I'm stuck solving a diophantine equation. It's mostly because I don't know how to solve equations resembling Pell's equation.

I have to solve this equation with (x,y) in Z.

x^2 + 2x + 3 = 10 y^2 + 11y + 12

I reduce the problem, into this :

(20y+11)^2 - 40*(x+1)^2 + 279 = 0

Y^2 - 40X^2 + 279 =0 with Y = 20y+11 and X=x+1

From that moment on, I don't know how to go on. Already, there is a condition where Y = 11 mod 20 and moreover I don't know how to solve it with Pell's equation.

Thanks for the help !