First, let's rearrange the equation by completing the square for y and x.
(y-3)^2 -9 + 2(x+2)^2 -8 = 367
Next, simplify and isolate the squared terms.
2(x+2)^2 + (y-3)^2 = 384
Divide both sides by 2 to obtain
(x+2)^2 + (y-3)^2 / 2= 192
To find integer solutions, we can try different values of x and solve for y. If y turns out to be an integer, then we have found a solution. It might also be helpful to note that the sum of two squares is an integer only if both squares are integer (or zero).
Let's try x = 0, 1, 2.
For x = 0, we have
(y-3)^2 = 384
This has no integer solution.
For x = 1, we have
(y-3)^2 = 382
This also has no integer solution.
For x = 2, we have
(y-3)^2 = 378
This gives us two integer solutions:
y-3 = ±6√3
y = 3 ± 6√3
So, the equation has two integer solutions:
(2, 3 + 6√3) and (2, 3 − 6√3)