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!

Use the completing the square technique to get $(y-3)^2+2(x+2)^2 = 384$. 
Make the substitution $a = y-3$ and $b = x+2$ to get $a^2+2b^2 = 384$. 
Since $2b^2$ and $384$ are even, $a^2$ must be even, and hence $a$ is even. So let $a = 2a_1$. 
Then, we get $4a_1^2+2b^2 = 384$, i.e. $2a_1^2+b^2 = 192$. 
Since $2a_1^2$ and $192$ are even, $b^2$ must be even, and hence $b$ is even. So let $b = 2b_1$. 
You can keep doing this until you get down to $2a_4^2+b_3^2 = 3$. 
Then, its easy to find all solutions and undo all the substitutions. 
