1. **State the problem:** Simplify the expression $$\frac{x^2 - 12xy + 20y^2}{x^2 - 4y^2} \div \frac{3x - 6y}{x + 2y}$$. 2. **Rewrite division as multiplication:** Dividing by a fraction is the same as multiplying by its reciprocal. $$\frac{x^2 - 12xy + 20y^2}{x^2 - 4y^2} \times \frac{x + 2y}{3x - 6y}$$ 3. **Factor all polynomials:** - Numerator of first fraction: $$x^2 - 12xy + 20y^2 = (x - 10y)(x - 2y)$$ because $-10y - 2y = -12y$ and $-10y \times -2y = 20y^2$. - Denominator of first fraction: $$x^2 - 4y^2 = (x - 2y)(x + 2y)$$ (difference of squares). - Denominator of second fraction: $$3x - 6y = 3(x - 2y)$$. 4. **Substitute factored forms:** $$\frac{(x - 10y)(x - 2y)}{(x - 2y)(x + 2y)} \times \frac{x + 2y}{3(x - 2y)}$$ 5. **Cancel common factors:** - Cancel $(x - 2y)$ from numerator and denominator. - Cancel $(x + 2y)$ from numerator and denominator. Remaining expression: $$\frac{x - 10y}{1} \times \frac{1}{3(x - 2y)} = \frac{x - 10y}{3(x - 2y)}$$ 6. **Final simplified answer:** $$\boxed{\frac{x - 10y}{3(x - 2y)}}$$ This is the simplest form because no further factoring or cancellation is possible.