jmc

For this problem, we use the fact that the sum of the roots of the polynomial $a{x}^2+bx+c$ is $-b/a$ and the product of the roots is $c/a.$

The positive factors of $28$ are $(1,28),$ $(2,14),$ $(4,7).$ Each of these sums are distinct. Hence, there are $\boxed{3}$ possible values for $m.$