Problem: Find $3 x^2 y^2$ if $x$ and $y$ are integers such that $y^2+3 x^2 y^2=30 x^2+517$. Solution: Notice that $(y^2-10)(3x^2+1) = 507$. From here, we find that $507$ is divisible by $1, 3, 13, 39, 169, 507$. The number $507$ can be factored as $3 \cdot 13^2$. Since $x$ and $y$ are integers, the expression $3x^2 + 1$ cannot be a multiple of three. Additionally, $169$ does not satisfy the equation, so we have $3x^2 + 1 = 13$, which leads to $x^2 = 4$. This results in $y^2 - 10 = 39$, giving $y^2 = 49$. Thus, $3x^2 y^2 = 3 \cdot 4 \cdot 49 = 588.$.