If both the roots of the quadratic equation $x^2-2px+p^2+p-5=0$ are less than 3, then k lies in the interval

For the equation x^2 - 2px + (p^2 + p - 5) = 0 to have both roots less than 3, three conditions must be met: the discriminant must be non-negative, the axis of symmetry must be less than 3, and the value of the quadratic at x = 3 must be positive. The discriminant condition is (-2p)^2 - 4(p^2 + p - 5) >= 0, which simplifies to 20 - 4p >= 0, meaning p <= 5. The axis of symmetry condition is p < 3. The value at x = 3 is 3^2 - 2p(3) + p^2 + p - 5 > 0, simplifying to p^2 - 5p + 4 > 0, which holds when p < 1 or p > 4. Intersecting the intervals p <= 5, p < 3, and (p < 1 or p > 4) results in the valid range (-infinity, 1).

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