For how many values of p, the circle x² + y² + 2x + 4y – p = 0 and the coordinate axes have exactly three common points?

The centre of the given circle is (-1, -2). For exactly three intercepts with the coordinate axes, there are three possibilities:

1. Tangent to Y-axis and intersecting X-axis in 2 points.
2. Tangent to X-axis and intersecting Y-axis in 2 points.
3. Passing through origin and intersecting the coordinate axes in one point each.
   In this case, if we take the circle tangent to Y-axis, it will not intersect the X-axis in any point. So, case 1 is dismissed.
   Hence, there are 2 possibilities as shown in the given figure.
   Hence, p can take 2 values only.