In order to have three real roots, the function must have a local minimum and a local maximum. For that to happen, the derivative must be zero at those points.
The derivative of x^3 + 2px^2 + 2p^2 x + p is
3x^2 + 4px + 2p^2
To solve this for zero, the discriminant is 16p^2 - 24p^2, which is negative, indicating no real roots. Therefore the derivative is never zero, there are no local minumum nor maximum, and the cubic has only one real root.
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