If $a$ and $b$ are distinct zeroes of the polynomial $x^3 - 2x + c$ and $a^2\left(2a^2 + 4ab + 3b^2\right) = 3$ then $b^2\left(3a^2 + 4ab + 2b^2\right)$ is equal to:
Step-by-Step Solution
Key Concept: Clever substitution can transform a complex system into a simpler algebraic relationship.
Starting with $(a^2 - 2a + c) - (b^2 - 2b + c) = 0$, we simplify to $a^2 + ab + b^2 = 2$. Using substitution $x = a^2(2a^2 + 4ab + 3b^2)$ and $y = b^2(3a^2 + 4ab + 2b^2)$, we find $x + y - 8 = 2(ab + 2)(a^2 + ab + b^2 - 2) = 0$, leading to $3 + y - 8 = 0$.
Correct Answer: 3