Example 4 Given that $x, y$ are integers, and
$$
15 x^{2} y^{2}=35 x^{2}-3 y^{2}+412 \text {. }
$$

then $15 x^{2} y^{2}=$ $\qquad$ .
Solution: Transform the given equation into
$$
\left(5 x^{2}+1\right)\left(3 y^{2}-7\right)=405=3^{4} \times 5 \text {. }
$$

Since $5 x^{2}+1$ is not a multiple of 5, and $3 y^{2}-7$ is not a multiple of 3, it can only be
$$
5 x^{2}+1=3^{4}, 3 y^{2}-7=5 \text {. }
$$

Therefore, $5 x^{2}=80,3 y^{2}=12$.
Thus, $15 x^{2} y^{2}=80 \times 12=960$.