1. **State the problem:** Simplify the expression $$\frac{2m^2 + 9m - 5}{3m^2 + 13m - 10} \div \frac{14m - 7}{4 - 9m^2}$$. 2. **Recall the division of fractions rule:** Dividing by a fraction is the same as multiplying by its reciprocal. 3. **Rewrite the division as multiplication:** $$\frac{2m^2 + 9m - 5}{3m^2 + 13m - 10} \times \frac{4 - 9m^2}{14m - 7}$$ 4. **Factor all polynomials where possible:** - Numerator 1: $2m^2 + 9m - 5 = (2m - 1)(m + 5)$ - Denominator 1: $3m^2 + 13m - 10 = (3m - 2)(m + 5)$ - Numerator 2: $4 - 9m^2 = (2 - 3m)(2 + 3m)$ (difference of squares) - Denominator 2: $14m - 7 = 7(2m - 1)$ 5. **Substitute the factored forms:** $$\frac{(2m - 1)(m + 5)}{(3m - 2)(m + 5)} \times \frac{(2 - 3m)(2 + 3m)}{7(2m - 1)}$$ 6. **Cancel common factors:** - Cancel $(m + 5)$ from numerator and denominator. - Cancel $(2m - 1)$ from numerator and denominator. Intermediate step showing cancellation: $$\frac{\cancel{(2m - 1)}\cancel{(m + 5)}}{(3m - 2)\cancel{(m + 5)}} \times \frac{(2 - 3m)(2 + 3m)}{7\cancel{(2m - 1)}}$$ 7. **Simplify the expression:** $$\frac{1}{3m - 2} \times \frac{(2 - 3m)(2 + 3m)}{7} = \frac{(2 - 3m)(2 + 3m)}{7(3m - 2)}$$ 8. **Note that $2 - 3m = -(3m - 2)$, so:** $$\frac{-(3m - 2)(2 + 3m)}{7(3m - 2)}$$ 9. **Cancel $(3m - 2)$ terms:** $$\frac{\cancel{-(3m - 2)}(2 + 3m)}{7\cancel{(3m - 2)}} = \frac{-(2 + 3m)}{7} = \frac{-2 - 3m}{7}$$ **Final answer:** $$\boxed{\frac{-2 - 3m}{7}}$$