1. **State the problem:** Solve the equation $$\frac{\left(2x - \frac{1}{3}\right)(3 - x)}{5} + \frac{(2x + 1)^2}{10} + \left(\frac{1}{10} - \frac{1}{5}\right)^{-1} \left(\frac{x}{10} + 5^{-1}\right) = 0$$ 2. **Simplify the inverse term:** Calculate $$\left(\frac{1}{10} - \frac{1}{5}\right)^{-1}$$ $$\frac{1}{10} - \frac{1}{5} = \frac{1}{10} - \frac{2}{10} = -\frac{1}{10}$$ So, $$\left(-\frac{1}{10}\right)^{-1} = -10$$ 3. **Rewrite the equation substituting the inverse:** $$\frac{\left(2x - \frac{1}{3}\right)(3 - x)}{5} + \frac{(2x + 1)^2}{10} - 10 \left(\frac{x}{10} + \frac{1}{5}\right) = 0$$ 4. **Simplify inside the parentheses:** $$\frac{x}{10} + \frac{1}{5} = \frac{x}{10} + \frac{2}{10} = \frac{x + 2}{10}$$ Multiply by -10: $$-10 \times \frac{x + 2}{10} = - (x + 2)$$ 5. **Rewrite the equation:** $$\frac{\left(2x - \frac{1}{3}\right)(3 - x)}{5} + \frac{(2x + 1)^2}{10} - (x + 2) = 0$$ 6. **Multiply entire equation by 10 to clear denominators:** $$10 \times \left[ \frac{\left(2x - \frac{1}{3}\right)(3 - x)}{5} + \frac{(2x + 1)^2}{10} - (x + 2) \right] = 10 \times 0$$ $$2 \times \left(2x - \frac{1}{3}\right)(3 - x) + (2x + 1)^2 - 10(x + 2) = 0$$ 7. **Expand terms:** First expand $$\left(2x - \frac{1}{3}\right)(3 - x)$$: $$= 2x \times 3 - 2x \times x - \frac{1}{3} \times 3 + \frac{1}{3} \times x = 6x - 2x^2 - 1 + \frac{x}{3}$$ Multiply by 2: $$2(6x - 2x^2 - 1 + \frac{x}{3}) = 12x - 4x^2 - 2 + \frac{2x}{3}$$ 8. **Expand $$(2x + 1)^2$$:** $$(2x + 1)^2 = 4x^2 + 4x + 1$$ 9. **Rewrite the equation:** $$12x - 4x^2 - 2 + \frac{2x}{3} + 4x^2 + 4x + 1 - 10x - 20 = 0$$ 10. **Combine like terms:** - Quadratic terms: $$-4x^2 + 4x^2 = 0$$ - Linear terms: $$12x + \frac{2x}{3} + 4x - 10x = (12x + 4x - 10x) + \frac{2x}{3} = 6x + \frac{2x}{3} = \frac{18x}{3} + \frac{2x}{3} = \frac{20x}{3}$$ - Constants: $$-2 + 1 - 20 = -21$$ 11. **Final simplified equation:** $$\frac{20x}{3} - 21 = 0$$ 12. **Solve for $x$:** $$\frac{20x}{3} = 21$$ Multiply both sides by 3: $$20x = 63$$ Divide both sides by 20: $$x = \frac{63}{20}$$ 13. **Final answer:** $$\boxed{\frac{63}{20}}$$