jmc

By considering the expression $\frac{-b\pm \sqrt{b^2-4ac}}{2a}$ for the roots of $a{x}^2+bx+c$, we find that the roots are real and distinct if and only if the discriminant $b^2-4ac$ is positive. So the roots of $x^2+mx+4$ are real and positive when $m^2-4(1)(4)>0$. Simplifying and factoring the left-hand side, we find $(m-4)(m+4)>0$, which implies $m\in \boxed{(-\infty ,-4)\cup (4,\infty )}$.