(b) \( 6 x^{2}-2 x^{3} \geq-x^{2}+5 x+1 \)

To tackle the inequality \( 6x^2 - 2x^3 \geq -x^2 + 5x + 1 \), let's first rearrange it by moving all terms to one side. This gives us: \[ -2x^3 + 7x^2 - 5x - 1 \geq 0 \] Now, factorizing or applying numerical methods can help find the roots of the polynomial \( -2x^3 + 7x^2 - 5x - 1 \). You may find critical points using techniques like synthetic division or the Rational Root Theorem! Next, test intervals around these roots to determine where the polynomial is positive or negative, giving you the solution to the inequality. It's a journey into the world of polynomials and their whimsical ways! Don't forget to visually explore the graph of your cubic polynomial for an exciting twist; spotting the crossing points with the x-axis can give you clues on the intervals that satisfy \( -2x^3 + 7x^2 - 5x - 1 \geq 0\). Now get out there and show those inequalities who’s boss!