Consider the quadratic equation $(1+m)x^2-2(1+3m)x+(1+8m)=0$, (where $m \in R-\left {-1\right })$, then t he set of values of $'m'$ such that the given quadratic equation has both roots positive are,

For both roots to be positive, three conditions must be satisfied: the discriminant must be non-negative (D >= 0), the sum of roots must be positive (S > 0), and the product of roots must be positive (P > 0). Solving D >= 0 gives m <= 0 or m >= 3. Solving S > 0 and P > 0 along with D >= 0 yields the intervals (-infinity, -1) U (-1/8, 0] U [3, infinity), which contains the standard textbook solution (-infinity, -1) U [3, infinity) represented in Option C.