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(D) For circle $C_1: x^2 + y^2 - 6x = 0$, center $C_1 = (3, 0)$ and radius $r_1 = \sqrt{3^2 + 0^2 - 0} = 3$.For circle $C_2: x^2 + y^2 + 6x + 2y + 1 =

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$4$

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(D) For circle $C_1: x^2 + y^2 - 6x = 0$, center $C_1 = (3, 0)$ and radius $r_1 = \sqrt{3^2 + 0^2 - 0} = 3$.For circle $C_2: x^2 + y^2 + 6x + 2y + 1 = 0$, center $C_2 = (-3, -1)$ and radius $r_2 = \sqrt{(-3)^2 + (-1)^2 - 1} = \sqrt{9 + 1 - 1} = 3$.The distance between the centers $d = \sqrt{(3 - (-3))^2 + (0 - (-1))^2} = \sqrt{6^2 + 1^2} = \sqrt{37}$.Since $r_1 + r_2 = 3 + 3 = 6$ and $\sqrt{37} > 6$, the distance between the centers is greater than the sum of the radii $(d > r_1 + r_2)$.Therefore, the circles are separate and do not intersect or touch each other.In this case, the number of common tangents is $4$.

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

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