All the roots of $x^3 + ax^2 + bx + c$ are positive integers greater than 2 and the coefficient satisfy $a + b + c = -46$:
Step-by-Step Solution
Key Concept: Transforming a sum condition on roots into a product form via Vieta's formulas enables factorization to find individual roots.
For a cubic with roots $\alpha, \beta, \gamma$, use Vieta's formulas: $\sum\alpha = -a$, $\sum\alpha\beta = b$, $\alpha\beta\gamma = -c$. The condition $\sum(a+b+c+1) = 45$ becomes $(\alpha-1)(\beta-1)(\gamma-1) = 3\times 3\times 5$, yielding $\alpha = 4, \beta = 4, \gamma = 6$, so $a = 4, b = 4, c = 6$ (taking the most natural factorization).
Correct Answer: 1,4