12.38 $x, y$ are integers that satisfy the equation $y^{2}+3 x^{2} y^{2}=30 x^{2}+517$. Find the value of $3 x^{2} y^{2}$.
(5th American Invitational Mathematics Examination, 1987)
[Solution] From $y^{2}+3 x^{2} y^{2}=30 x^{2}+517$
we get
$$
\begin{array}{c}
y^{2}+3 x^{2} y^{2}-30 x^{2}-10=507, \\
\left(y^{2}-10\right)\left(3 x^{2}+1\right)=3 \cdot 169=3 \cdot 13^{2} .
\end{array}
$$

Since $x, y$ are integers, we have
$$
\begin{array}{c}
y^{2}-10=1,3,13,39,169,507, \\
y^{2}=11,13,23,49,179,517 .
\end{array}
$$

Among these, only 49 is a perfect square, so
$$
y^{2}=49, \quad y^{2}-10=39 \text {. }
$$

Thus,
$$
\begin{array}{c}
3 x^{2}+1=13, \\
x^{2}=4 .
\end{array}
$$

Therefore,
$$
3 x^{2} y^{2}=3 \cdot 4 \cdot 49=588
$$