If z is a complex number with |z-2-3i|=3, then the greatest possible value of |z+2| is

The condition says that $z$ lies on the circle with centre $C(2,3)$ and radius $3$. The quantity $|z+2|=|z-(-2)|$ is the distance of $z$ from the point $Q(-2,0)$. The distance from $Q$ to the centre is $|CQ|=\sqrt{(2+2)^2+(3-0)^2}=5$. Since $Q$ lies outside the circle ($5>3$), the greatest distance from $Q$ to a point of the circle is attained on the line $QC$ beyond the centre and equals $|CQ|+r=5+3=8$. The least value would be $5-3=2$.