1. **State the problem:** Multiply the rational expressions $$\frac{4x - 7y}{3x^{5} y^{7}} \cdot \frac{18x^{4} y^{9}}{12x - 21y}$$ and simplify the result. 2. **Rewrite the expression:** $$\frac{4x - 7y}{3x^{5} y^{7}} \times \frac{18x^{4} y^{9}}{12x - 21y}$$ 3. **Factor where possible:** - Factor the denominator of the second fraction: $$12x - 21y = 3(4x - 7y)$$ 4. **Substitute the factorization:** $$\frac{4x - 7y}{3x^{5} y^{7}} \times \frac{18x^{4} y^{9}}{3(4x - 7y)}$$ 5. **Multiply the numerators and denominators:** $$\frac{(4x - 7y) \times 18x^{4} y^{9}}{3x^{5} y^{7} \times 3(4x - 7y)}$$ 6. **Cancel common factors:** - Cancel $4x - 7y$ in numerator and denominator: $$\frac{\cancel{4x - 7y} \times 18x^{4} y^{9}}{3x^{5} y^{7} \times 3\cancel{(4x - 7y)}} = \frac{18x^{4} y^{9}}{9x^{5} y^{7}}$$ 7. **Simplify coefficients:** $$\frac{18}{9} = 2$$ 8. **Simplify variables using laws of exponents:** - For $x$: $$\frac{x^{4}}{x^{5}} = x^{4-5} = x^{-1} = \frac{1}{x}$$ - For $y$: $$\frac{y^{9}}{y^{7}} = y^{9-7} = y^{2}$$ 9. **Write the simplified expression:** $$2 \times \frac{1}{x} \times y^{2} = \frac{2y^{2}}{x}$$ **Final answer:** $$\boxed{\frac{2y^{2}}{x}}$$