The number of common tangents that can be drawn to the circles $x^2+y^2-6 x=0$ and $x^2+y^2+6 x+2 y+1=0$ is .....

Common tangents to two circles

The centers and radii of the circles are determined using the standard circle equation $x^2+y^2+2gx+2fy+c=0$.

For the first circle $x^2+y^2-6x=0$, the center is $C_1=(3,0)$ and radius $r_1=3$.

For the second circle $x^2+y^2+6x+2y+1=0$, the center is $C_2=(-3,-1)$ and radius $r_2=3$.

The distance between centers is $d=\sqrt{(-3-3)^2+(-1-0)^2}=\sqrt{37}$.

Since $d>r_1+r_2$ because $\sqrt{37}>6$, the circles lie completely outside each other.

Two non-intersecting circles have four common tangents: two direct and two transverse.

The number of common tangents is $\boxed{4}$.