Lost. Sorry.
$\displaystyle 4x^2 - y^2 = (2x + y)(2x - y) = 480$.
Now list all the pairs of numbers that can multiply to 480:
$\displaystyle (480, 1); (240, 2); (160, 3); (120, 4); (96, 5); (80, 6); $
$\displaystyle (60, 8); (48, 10); (40, 12); (32, 15); (30, 16); (24, 20)$
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. $\displaystyle 2x + y = 120$
$\displaystyle 2x - y = 4$
$\displaystyle 4x = 124$
$\displaystyle x = 31, y = 58$ is a solution.
2. $\displaystyle 2x + y = 60$
$\displaystyle 2x - y = 8$
$\displaystyle 4x = 68$
$\displaystyle x = 17, y = 26$ is a solution.
3. $\displaystyle 2x + y = 40$
$\displaystyle 2x - y = 12$
$\displaystyle 4x = 52$
$\displaystyle x = 13, y = 14$ is a solution.
4. $\displaystyle 2x + y = 24$
$\displaystyle 2x - y = 20$
$\displaystyle 4x = 44$
$\displaystyle x = 11, y = 2$ is a solution.
Hence the solutions are $\displaystyle (31, 58); (17, 26); (13, 14); (11, 2)$.