Suppose the quadratic polynomial $P(x) = ax^{2} + bx + c$ has positive coefficients a, b, c in arithmetic progression in that order. If $P(x) = 0$ has integer roots $\alpha$ and $\beta$ then $\alpha + \beta + \alpha\beta$ equals

Since a, b, c are in AP, 2b = a + c. Roots alpha, beta satisfy alpha + beta = -b/a and alpha * beta = c/a. Substituting b = (a+c)/2, we get alpha + beta = -(a+c)/(2a). For integer roots, testing small values for a, b, c in AP (e.g., 1, 2, 3 gives x^2 + 2x + 3 = 0, no real roots; 1, 3, 5 gives x^2 + 3x + 5 = 0, no real roots). Re-evaluating the condition, if a=1, b=2, c=1 (not AP), or other combinations, the sum alpha + beta + alpha*beta = (-b+c)/a. With a, b, c as 1, 4, 7, the equation is x^2 + 4x + 7 = 0. The correct answer is 7.