1. The problem states the cubic equation $$4x^3 + 3x^2 + 2x = 12$$ and mentions finding its roots. 2. The solution approximates the only real root as $$x \approx 1.136$$ by testing rational roots and using numerical methods. 3. However, the solution is missing the explicit steps showing how to test for rational roots, such as using the Rational Root Theorem. 4. It also lacks the detailed explanation or demonstration of the numerical method used (e.g., Newton-Raphson, bisection) to approximate the root between $$x=1.135$$ and $$x=1.136$$. 5. The solution mentions the discriminant confirming two complex roots but does not show the calculation or formula for the discriminant of a cubic equation. 6. Including these missing details would provide a complete understanding of how the root approximation and root nature were determined. Final answer: The missing information includes the rational root testing process, the numerical method steps for approximation, and the discriminant calculation to confirm complex roots.