If -1 is one of the roots of the cubic equation x³ + (3s - 2)x² + (3s + 1)x + 4 = 0, and the other two roots of equation are real, then the minimum positive integral value of s is:

Since -1 is a root, (-1)^3 + (3s-2)(-1)^2 + (3s+1)(-1) + 4 = 0. -1 + 3s - 2 - 3s - 1 + 4 = 0. This simplifies to 0=0, which is always true for any s. For the other two roots to be real, the discriminant of the quadratic factor must be >= 0. Dividing the cubic by (x+1) gives x^2 + (3s-3)x + 4 = 0. Discriminant D = (3s-3)^2 - 16 >= 0. (3s-3)^2 >= 16. 3s-3 >= 4 or 3s-3 <= -4. 3s >= 7 (s >= 2.33) or 3s <= -1. Smallest positive integer s is 3.