In a quadratic function Z(x) = px² + mx + n, where p, m, and n are integers, if Z(5) = Z(-2) = 0 and p(≠ 1) is a positive integer, then which of the following is the maximum value of m + p?

Since Z(5) = Z(-2) = 0, the roots are 5 and -2. The function is Z(x) = p(x - 5)(x + 2) = p(x^2 - 3x - 10) = px^2 - 3px - 10p. Comparing to px^2 + mx + n, we get m = -3p and n = -10p. We want to maximize m + p = -3p + p = -2p. Since p is a positive integer and p != 1, the smallest p is 2. For p = 2, m + p = -4. For larger p, the value becomes more negative.