Answer
$-6+\sqrt{38}$
Work Step by Step
The center of this sphere is $(1,-2,0)$
Distance between $(4,0,5)$ and $(1,-2,0)$ is
\[
\sqrt{(-0-2)^{2}+(-5+0)^{2}+(-4+1)^{2}}=\sqrt{(-5)^{2}+(-3)^{2}+(-2)^{2}}=\sqrt{25+9+4}=\sqrt{38}
\]
Notice that radius of this sphere is $6,$ and $\sqrt{38}>6$
Thus, the point $(4,0,5)$ is outside the sphere
And, therefore, the shortest distance to the sphere is $-6+\sqrt{38}$.
