1. **State the problem:** Simplify the expression $$\frac{18x - 200x^3}{40xy^2 - 12y^2}$$. 2. **Identify the formula and rules:** To simplify a fraction, factor numerator and denominator and cancel common factors. 3. **Factor the numerator:** $$18x - 200x^3 = 2x(9 - 100x^2)$$ 4. **Factor the denominator:** $$40xy^2 - 12y^2 = 4y^2(10x - 3)$$ 5. **Rewrite the expression:** $$\frac{2x(9 - 100x^2)}{4y^2(10x - 3)}$$ 6. **Check for common factors:** Note that $$9 - 100x^2 = (3)^2 - (10x)^2 = (3 - 10x)(3 + 10x)$$. 7. **Rewrite numerator with difference of squares:** $$2x(3 - 10x)(3 + 10x)$$ 8. **Rewrite denominator:** $$4y^2(10x - 3)$$ 9. **Notice that $$10x - 3 = -(3 - 10x)$$, so we can write denominator as $$4y^2(-(3 - 10x)) = -4y^2(3 - 10x)$$. 10. **Cancel common factor $$(3 - 10x)$$:** $$\frac{2x(3 + 10x)}{-4y^2} = -\frac{2x(3 + 10x)}{4y^2}$$ 11. **Simplify the fraction:** $$-\frac{2x(3 + 10x)}{4y^2} = -\frac{x(3 + 10x)}{2y^2}$$ **Final answer:** $$-\frac{x(3 + 10x)}{2y^2}$$