1. **State the problem:** Simplify the expression $$\frac{\sqrt[3]{-125} \times (1 - 0.8)^2 + 0.3}{(\frac{1}{5})^{-1} - \sqrt{2.25}}$$. 2. **Recall formulas and rules:** - Cube root: $$\sqrt[3]{a}$$ is the number that when cubed gives $$a$$. - Powers and roots: $$a^{-n} = \frac{1}{a^n}$$. - Square root: $$\sqrt{a}$$ is the number that when squared gives $$a$$. - Order of operations: parentheses, exponents, multiplication/division, addition/subtraction. 3. **Calculate cube root:** $$\sqrt[3]{-125} = -5$$ because $$(-5)^3 = -125$$. 4. **Calculate inside parentheses:** $$1 - 0.8 = 0.2$$. 5. **Square the result:** $$(0.2)^2 = 0.04$$. 6. **Multiply cube root by squared term:** $$-5 \times 0.04 = -0.2$$. 7. **Add 0.3:** $$-0.2 + 0.3 = 0.1$$. 8. **Calculate denominator:** - $$\left(\frac{1}{5}\right)^{-1} = 5$$ because negative exponent inverts the fraction. - $$\sqrt{2.25} = 1.5$$ because $$1.5^2 = 2.25$$. 9. **Subtract in denominator:** $$5 - 1.5 = 3.5$$. 10. **Form the fraction:** $$\frac{0.1}{3.5}$$. 11. **Simplify fraction:** $$\frac{0.1}{3.5} = \frac{\cancel{0.1}}{\cancel{3.5}} = \frac{1}{35}$$ after multiplying numerator and denominator by 10. 12. **Final answer:** $$\boxed{\frac{1}{35}}$$.