Number of Solutions of $y^2-6y+2x^2+8x=367$? Find the number of solutions in integers to the equation 
$$y^2-6y+2x^2+8x=367$$
How should I go about solving this?
Thanks!

Note: this  method requires basic knowledge of conic sections.
First of all it is equation of an ellipse,

And its vertex are $(-2,3-8\sqrt6)|(-2,3+8\sqrt6) \approx(-2,-16.6)|(-2,22.6)$
So, you can vary $y$ from $-16$ to $22$ and you will get all integral solutions.
