Answer 1

If we move the x² term to the left side, it is factorable:
(3x² + 1) (y² – 10) = 517 – 10 = 507 is equal to 3 x 13².
Since x and y are integers, 3x² + 1 cannot equal a multiple of 3.
169 doesn't work either, so 3x² + 1 = 13, and x = ± 2.
This leaves y² – 10 = 39, so y = ± 7.
Thus, 3x²y² = 3 x 4 x 49 = 588.