If all roots of the polynomials $6x^2 - 24x - 4a$ and $x^3 + ax^2 + bx - 8$ are non-negative real numbers, then:
Step-by-Step Solution
Key Concept: Applying AM-GM to the roots combined with the requirement that the derivative quadratic has real roots uniquely determines the cubic.
Let $\alpha, \beta, \gamma$ be the roots of a cubic. The AM-GM inequality condition $AM \geq GM$ gives $-a \geq 6$, and examining $6x^2 - 24x - 4a = 0$ for all real roots requires $24^2 + 96a \geq 0$, yielding $a = -6$. Thus $\alpha = \beta = \gamma = 2$.
Correct Answer: 1,4