4. $2 \sqrt{10}$.

Let $x=X+3, y=Y+1$.
Then $21 X^{2}-44 X Y-96 Y^{2}=200$.
Let $X=\frac{11 \sqrt{5}}{25} a+\frac{2 \sqrt{5}}{25} b$,
$Y=-\frac{2 \sqrt{5}}{25} a+\frac{11 \sqrt{5}}{25} b$.
Thus, $\frac{a^{2}}{8}-\frac{b^{2}}{2}=1$.
Therefore, the distance between the foci is $2 \sqrt{10}$.
4. The distance between the foci of the conic section
$$
(3 x+4 y-13)(7 x-24 y+3)=200
$$

is $\qquad$