Diophantine Equation? Or something?

$4x^2 - y^2 = (2x + y)(2x - y) = 480$ .

Now list all the pairs of numbers that can multiply to 480:

$(480, 1); (240, 2); (160, 3); (120, 4); (96, 5); (80, 6);$

$(60, 8); (48, 10); (40, 12); (32, 15); (30, 16); (24, 20)$

The pairs with one odd number and one even number will result in y not being an integer, so you can throw those out. Also, the pairs such that the sum of the numbers is not divisible by 4 will result in x not being an integer, so you can throw those out.

1. $2x + y = 120$

$2x - y = 4$

$4x = 124$

$x = 31, y = 58$ is a solution.

2. $2x + y = 60$

$2x - y = 8$

$4x = 68$

$x = 17, y = 26$ is a solution.

3. $2x + y = 40$

$2x - y = 12$

$4x = 52$

$x = 13, y = 14$ is a solution.

4. $2x + y = 24$

$2x - y = 20$

$4x = 44$

$x = 11, y = 2$ is a solution.

Hence the solutions are $(31, 58); (17, 26); (13, 14); (11, 2)$ .