1. **State the problem:** Simplify the expression $$\frac{x^2 - 9}{x^2 - 7x + 12} \div \frac{x}{5x - 20}$$ as a single fraction in fully factorised form. 2. **Rewrite division as multiplication:** Dividing by a fraction is the same as multiplying by its reciprocal. $$\frac{x^2 - 9}{x^2 - 7x + 12} \times \frac{5x - 20}{x}$$ 3. **Factorise all polynomials:** - $x^2 - 9$ is a difference of squares: $$x^2 - 9 = (x - 3)(x + 3)$$ - $x^2 - 7x + 12$ factors as: $$x^2 - 7x + 12 = (x - 3)(x - 4)$$ - $5x - 20$ factors out 5: $$5x - 20 = 5(x - 4)$$ 4. **Substitute the factors back:** $$\frac{(x - 3)(x + 3)}{(x - 3)(x - 4)} \times \frac{5(x - 4)}{x}$$ 5. **Cancel common factors:** - Cancel $(x - 3)$ from numerator and denominator. - Cancel $(x - 4)$ from numerator and denominator. Intermediate step showing cancellation: $$\frac{\cancel{(x - 3)}(x + 3)}{\cancel{(x - 3)}\cancel{(x - 4)}} \times \frac{5\cancel{(x - 4)}}{x} = \frac{(x + 3)}{1} \times \frac{5}{x}$$ 6. **Multiply the remaining factors:** $$\frac{(x + 3) \times 5}{x} = \frac{5(x + 3)}{x}$$ 7. **Final answer:** $$\boxed{\frac{5(x + 3)}{x}}$$ This is the expression in its simplest, fully factorised form.