Which of the following points is farthest from the origin: $(1,4)$, $(3,5)$, $(-6,0)$, $(-4,-2)?$
We find the distance for all the points using the distance formula:
For $(1,4)$: $\sqrt{(1-0)^2+(4-0)^2}=\sqrt{17}$
For $(3,5)$: $\sqrt{(3-0)^2+(5-0)^2}=\sqrt{34}$
For $(-6,0)$: $\sqrt{(-6-0)^2+(0-0)^2}=\sqrt{36}$
For $(-4,-2)$: $\sqrt{(-4-0)^2+(-2-0)^2}=\sqrt{20}$
Therefore, the point farthest from the origin is $\boxed{(-6,0)}$.