1. **State the problem:** Solve the inequality $$\frac{3x^2}{5} + \frac{x}{4} - (x - 2) - \frac{1}{2}\left(x - \frac{1}{2}\right) \geq \left(\frac{x - 1}{2}\right)^2 + \frac{3x}{5}(x - 1).$$ 2. **Rewrite the inequality clearly:** $$\frac{3x^2}{5} + \frac{x}{4} - (x - 2) - \frac{1}{2}\left(x - \frac{1}{2}\right) \geq \left(\frac{x - 1}{2}\right)^2 + \frac{3x}{5}(x - 1).$$ 3. **Expand and simplify each term:** - Left side: $$\frac{3x^2}{5} + \frac{x}{4} - x + 2 - \frac{1}{2}x + \frac{1}{4}$$ Combine like terms: $$\frac{3x^2}{5} + \left(\frac{x}{4} - x - \frac{1}{2}x\right) + \left(2 + \frac{1}{4}\right) = \frac{3x^2}{5} + \left(\frac{x}{4} - \frac{3x}{2}\right) + \frac{9}{4}.$$ 4. **Simplify the x terms:** $$\frac{x}{4} - \frac{3x}{2} = \frac{x}{4} - \frac{6x}{4} = -\frac{5x}{4}.$$ So left side is: $$\frac{3x^2}{5} - \frac{5x}{4} + \frac{9}{4}.$$ 5. **Right side:** $$\left(\frac{x - 1}{2}\right)^2 + \frac{3x}{5}(x - 1) = \frac{(x - 1)^2}{4} + \frac{3x(x - 1)}{5}.$$ Expand: $$\frac{x^2 - 2x + 1}{4} + \frac{3x^2 - 3x}{5}.$$ 6. **Find common denominators and combine right side:** Multiply first term by 5/5 and second by 4/4: $$\frac{5(x^2 - 2x + 1)}{20} + \frac{4(3x^2 - 3x)}{20} = \frac{5x^2 - 10x + 5 + 12x^2 - 12x}{20} = \frac{17x^2 - 22x + 5}{20}.$$ 7. **Rewrite inequality:** $$\frac{3x^2}{5} - \frac{5x}{4} + \frac{9}{4} \geq \frac{17x^2 - 22x + 5}{20}.$$ 8. **Multiply both sides by 20 to clear denominators:** $$20 \times \left(\frac{3x^2}{5} - \frac{5x}{4} + \frac{9}{4}\right) \geq 17x^2 - 22x + 5.$$ Calculate left side: $$20 \times \frac{3x^2}{5} = 12x^2,$$ $$20 \times -\frac{5x}{4} = -25x,$$ $$20 \times \frac{9}{4} = 45.$$ So: $$12x^2 - 25x + 45 \geq 17x^2 - 22x + 5.$$ 9. **Bring all terms to one side:** $$12x^2 - 25x + 45 - 17x^2 + 22x - 5 \geq 0,$$ Simplify: $$-5x^2 - 3x + 40 \geq 0.$$ 10. **Multiply both sides by -1 (remember to flip inequality):** $$5x^2 + 3x - 40 \leq 0.$$ 11. **Solve quadratic inequality:** Find roots of $$5x^2 + 3x - 40 = 0.$$ Use quadratic formula: $$x = \frac{-3 \pm \sqrt{3^2 - 4 \times 5 \times (-40)}}{2 \times 5} = \frac{-3 \pm \sqrt{9 + 800}}{10} = \frac{-3 \pm \sqrt{809}}{10}.$$ 12. **Approximate roots:** $$\sqrt{809} \approx 28.44,$$ So roots are approximately: $$x_1 = \frac{-3 - 28.44}{10} = -3.144,$$ $$x_2 = \frac{-3 + 28.44}{10} = 2.544.$$ 13. **Determine solution interval:** Since leading coefficient $5 > 0$, parabola opens upward, so inequality $\leq 0$ holds between roots: $$-3.144 \leq x \leq 2.544.$$ **Final answer:** $$\boxed{-\frac{3 + \sqrt{809}}{10} \leq x \leq \frac{-3 + \sqrt{809}}{10}}.$$