To solve this problem, we analyze the quadratic function and its properties:

Step 1: Identify the nature of the quadratic function

The given function is (f(x) = x^2 + ax + b). Since the coefficient of (x^2) is positive ((1)), the parabola opens upwards.

Step 2: Relate roots to function values

A quadratic function opening upwards has the following behavior:

• It is negative between its two distinct real roots.
• It is positive outside the interval formed by the roots.

Step 3: Apply the given condition

The roots (p) and (q) satisfy (p < 2 < q), meaning (2) lies between the roots.

Since the function is negative between the roots, (f(2) < 0).

Answer: A

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