Given:
The equation of the circle is: x^2 + y^2 + 2x + 4y + p = 0

To find:
The number of values of p for which the circle and the coordinate axes have exactly three common points.

Solution:

To find the common points between the circle and the coordinate axes, we substitute the following values of y and x in the equation of the circle:

1. When y = 0, substituting y = 0 in the equation of the circle, we get:
x^2 + 2x + p = 0
This is a quadratic equation in x. For exactly three common points, the quadratic equation should have two real and distinct solutions. Therefore, the discriminant of the equation should be greater than zero.
Discriminant (D) = b^2 - 4ac
Substituting the values a = 1, b = 2, and c = p in the discriminant formula:
D = 2^2 - 4(1)(p)
D = 4 - 4p
For D > 0, we have:
4 - 4p > 0
4p < />
p < />

2. When x = 0, substituting x = 0 in the equation of the circle, we get:
y^2 + 4y + p = 0
This is a quadratic equation in y. Again, for exactly three common points, the discriminant of the equation should be greater than zero.
Substituting the values a = 1, b = 4, and c = p in the discriminant formula:
D = 4^2 - 4(1)(p)
D = 16 - 4p
For D > 0, we have:
16 - 4p > 0
4p < />
p < />

Summary:
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The maximum value of p that satisfies both conditions is p = 1.
Therefore, there are 2 values of p (0 and 1) for which the circle and the coordinate axes have exactly three common points.
Hence, the correct answer is option C.