1. **State the problem:** Simplify the expression $$0.2 \times 0.3 \times \left(1 - \frac{1}{10}\right)^{-1} \times \sqrt[3]{\frac{98}{125} - 1} + \left(0.2 + \frac{0.5}{0.25}\right)^{-1}$$ 2. **Rewrite decimals as fractions:** $$0.2 = \frac{1}{5}, \quad 0.3 = \frac{3}{10}, \quad 0.5 = \frac{1}{2}, \quad 0.25 = \frac{1}{4}$$ 3. **Simplify inside the parentheses:** $$1 - \frac{1}{10} = \frac{10}{10} - \frac{1}{10} = \frac{9}{10}$$ 4. **Apply the negative exponent:** $$\left(\frac{9}{10}\right)^{-1} = \frac{10}{9}$$ 5. **Simplify inside the cube root:** $$\frac{98}{125} - 1 = \frac{98}{125} - \frac{125}{125} = -\frac{27}{125}$$ 6. **Calculate the cube root:** $$\sqrt[3]{-\frac{27}{125}} = -\sqrt[3]{\frac{27}{125}} = -\frac{3}{5}$$ 7. **Simplify the addition inside the inverse:** $$0.2 + \frac{0.5}{0.25} = \frac{1}{5} + \frac{\frac{1}{2}}{\frac{1}{4}} = \frac{1}{5} + 2 = \frac{1}{5} + \frac{10}{5} = \frac{11}{5}$$ 8. **Apply the negative exponent:** $$\left(\frac{11}{5}\right)^{-1} = \frac{5}{11}$$ 9. **Substitute all simplified parts back:** $$\frac{1}{5} \times \frac{3}{10} \times \frac{10}{9} \times \left(-\frac{3}{5}\right) + \frac{5}{11}$$ 10. **Multiply the fractions step-by-step:** $$\frac{1}{5} \times \frac{3}{10} = \frac{3}{50}$$ $$\frac{3}{50} \times \frac{10}{9} = \frac{3 \times 10}{50 \times 9} = \frac{30}{450}$$ Simplify numerator and denominator by 30: $$\frac{\cancel{30}^{1}}{\cancel{450}^{15}} = \frac{1}{15}$$ 11. **Multiply by $-\frac{3}{5}$:** $$\frac{1}{15} \times \left(-\frac{3}{5}\right) = -\frac{3}{75} = -\frac{1}{25}$$ 12. **Add $\frac{5}{11}$:** $$-\frac{1}{25} + \frac{5}{11} = \frac{-11}{275} + \frac{125}{275} = \frac{114}{275}$$ 13. **Final answer:** $$\boxed{\frac{114}{275}}$$