$\text{ If three exist a complex number }Z\text{ satisfying both }|Z-mi|=m+5\text{ and }|Z-4|<3\text{ , then}$

$\left|z-mi\right|=m+5 represents a circle having centre at mi and radius m+5$

$and \left|z-4\right|<3 is inside region of a circle having centre at \left(4,0\right) and radius 3$

$The two circles intersect$

$\begin{array}{l} therefore r_1-r_2<C_1{C}_2<r_1+r_2\\ \Rightarrow m+5-3<\sqrt{m^2+16}<m+5+3\\ \Rightarrow m\in (-3,3)\end{array}$