Final answer:
The student asked for a quadratic function with roots -2 and p. The correct quadratic function given these roots and a leading coefficient of 1 is f(x) = (x + 2)(x - p), so the correct option is a).
Step-by-step explanation:
The student has asked to write a quadratic function with a leading coefficient of 1 that has roots of -2 and p. The roots of a quadratic function are the values of x for which the function equals zero. In other words, if -2 and p are the roots of the function, then f(x) = 0 when x is either -2 or p. The general form of a quadratic function is f(x) = a(x - r1)(x - r2) where a is the leading coefficient, and r1 and r2 are the roots.
Given that the leading coefficient is 1, and one of the roots is -2, the correct quadratic function would be f(x) = (x + 2)(x - p) because:
- For x = -2, the first part of the product becomes zero, which satisfies the condition that -2 is a root.
- For x = p, the second part of the product becomes zero, which satisfies the condition that p is the other root.
The other options either do not satisfy the roots properly or change the sign of the roots incorrectly. Therefore, the correct option is a) f(x) = (x + 2)(x - p).