Find $3x^2 y^2$ if $x$ and $y$ are integers such that $y^2 + 3x^2 y^2 = 30x^2 + 517$.
1. Given the equation \( y^2 + 3x^2 y^2 = 30x^2 + 517 \), we start by letting \( a = x^2 \) and \( b = y^2 \). This transforms the equation into:
   \[
   b + 3ab = 30a + 517
   \]

2. We can use Simon's Favorite Factoring Trick to factor the equation. First, we rearrange the terms:
   \[
   b + 3ab - 30a = 517
   \]

3. Next, we factor out \( b \) on the left-hand side:
   \[
   b(1 + 3a) = 30a + 517
   \]

4. To make the equation easier to factor, we add and subtract 10 on the left-hand side:
   \[
   b(1 + 3a) - 10(1 + 3a) = 517 - 10(1 + 3a)
   \]

5. This simplifies to:
   \[
   (b - 10)(1 + 3a) = 507
   \]

6. We know that \( 507 = 3 \cdot 13^2 \). Therefore, we can write:
   \[
   (b - 10)(1 + 3a) = 3 \cdot 13^2
   \]

7. We now consider the factor pairs of 507:
   \[
   (1, 507), (3, 169), (13, 39), (39, 13), (169, 3), (507, 1)
   \]

8. We solve for \( a \) and \( b \) using these pairs:
   - For \( (1 + 3a) = 1 \) and \( (b - 10) = 507 \):
     \[
     1 + 3a = 1 \implies a = 0
     \]
     \[
     b - 10 = 507 \implies b = 517
     \]
     This gives \( (a, b) = (0, 517) \), but \( a = 0 \) is not a perfect square.

   - For \( (1 + 3a) = 3 \) and \( (b - 10) = 169 \):
     \[
     1 + 3a = 3 \implies 3a = 2 \implies a = \frac{2}{3}
     \]
     This is not an integer, so it is not valid.

   - For \( (1 + 3a) = 13 \) and \( (b - 10) = 39 \):
     \[
     1 + 3a = 13 \implies 3a = 12 \implies a = 4
     \]
     \[
     b - 10 = 39 \implies b = 49
     \]
     This gives \( (a, b) = (4, 49) \), which are both perfect squares.

   - For \( (1 + 3a) = 39 \) and \( (b - 10) = 13 \):
     \[
     1 + 3a = 39 \implies 3a = 38 \implies a = \frac{38}{3}
     \]
     This is not an integer, so it is not valid.

   - For \( (1 + 3a) = 169 \) and \( (b - 10) = 3 \):
     \[
     1 + 3a = 169 \implies 3a = 168 \implies a = 56
     \]
     \[
     b - 10 = 3 \implies b = 13
     \]
     This gives \( (a, b) = (56, 13) \), but \( b = 13 \) is not a perfect square.

   - For \( (1 + 3a) = 507 \) and \( (b - 10) = 1 \):
     \[
     1 + 3a = 507 \implies 3a = 506 \implies a = \frac{506}{3}
     \]
     This is not an integer, so it is not valid.

9. The only valid solution is \( (a, b) = (4, 49) \). Therefore, we have:
   \[
   x^2 = 4 \implies x = 2 \text{ or } x = -2
   \]
   \[
   y^2 = 49 \implies y = 7 \text{ or } y = -7
   \]

10. We need to find \( 3x^2 y^2 \):
    \[
    3x^2 y^2 = 3 \cdot 4 \cdot 49 = 3 \cdot 196 = 588
    \]

The final answer is \(\boxed{588}\)