Four, try to find such a positive integer $a$, so that the quadratic equation
$$
a x^{2}+2(2 a-1) x+4(a-3)=0
$$

has at least one integer root.
Given $-4 \leqslant x \leqslant 2$, by testing integer values of $x$, we can obtain $a=1, 3, 6, 10$.