1. **State the problem:** Simplify the expression $$\frac{3a + 2}{2a^2 + 11a + 5} - \frac{a - 2}{6a^2 - 7a - 5} \div \frac{2a}{3a^2 - 5a}$$. 2. **Factor all quadratic expressions:** - Factor $2a^2 + 11a + 5$: $$2a^2 + 11a + 5 = (2a + 1)(a + 5)$$ - Factor $6a^2 - 7a - 5$: $$6a^2 - 7a - 5 = (2a + 1)(3a - 5)$$ - Factor $3a^2 - 5a$: $$3a^2 - 5a = a(3a - 5)$$ 3. **Rewrite the expression with factored denominators:** $$\frac{3a + 2}{(2a + 1)(a + 5)} - \frac{a - 2}{(2a + 1)(3a - 5)} \div \frac{2a}{a(3a - 5)}$$ 4. **Rewrite division as multiplication by reciprocal:** $$\frac{3a + 2}{(2a + 1)(a + 5)} - \frac{a - 2}{(2a + 1)(3a - 5)} \times \frac{a(3a - 5)}{2a}$$ 5. **Simplify the multiplication:** $$\frac{a(3a - 5)}{2a} = \cancel{\frac{a}{2a}} (3a - 5) = \frac{1}{2}(3a - 5)$$ 6. **Substitute back:** $$\frac{3a + 2}{(2a + 1)(a + 5)} - \frac{a - 2}{(2a + 1)(3a - 5)} \times \frac{1}{2}(3a - 5)$$ 7. **Cancel $(3a - 5)$ in numerator and denominator:** $$\frac{3a + 2}{(2a + 1)(a + 5)} - \frac{a - 2}{(2a + 1)\cancel{(3a - 5)}} \times \frac{1}{2}\cancel{(3a - 5)} = \frac{3a + 2}{(2a + 1)(a + 5)} - \frac{a - 2}{(2a + 1)} \times \frac{1}{2}$$ 8. **Multiply the second term:** $$\frac{3a + 2}{(2a + 1)(a + 5)} - \frac{(a - 2)}{2(2a + 1)}$$ 9. **Find common denominator:** The denominators are $(2a + 1)(a + 5)$ and $2(2a + 1)$. The least common denominator (LCD) is $2(2a + 1)(a + 5)$. 10. **Rewrite each fraction with LCD:** $$\frac{3a + 2}{(2a + 1)(a + 5)} = \frac{2(3a + 2)}{2(2a + 1)(a + 5)}$$ $$\frac{(a - 2)}{2(2a + 1)} = \frac{(a - 2)(a + 5)}{2(2a + 1)(a + 5)}$$ 11. **Combine the fractions:** $$\frac{2(3a + 2) - (a - 2)(a + 5)}{2(2a + 1)(a + 5)}$$ 12. **Expand numerator:** - $2(3a + 2) = 6a + 4$ - $(a - 2)(a + 5) = a^2 + 5a - 2a - 10 = a^2 + 3a - 10$ 13. **Substitute and simplify numerator:** $$6a + 4 - (a^2 + 3a - 10) = 6a + 4 - a^2 - 3a + 10 = -a^2 + 3a + 14$$ 14. **Final simplified expression:** $$\frac{-a^2 + 3a + 14}{2(2a + 1)(a + 5)}$$ 15. **Optional: factor numerator if possible:** Check factors of $-a^2 + 3a + 14$: Rewrite as $-(a^2 - 3a - 14)$. Factors of $a^2 - 3a - 14$ are $(a - 7)(a + 2)$. So numerator: $$-(a - 7)(a + 2)$$ 16. **Write fully factored form:** $$\frac{-(a - 7)(a + 2)}{2(2a + 1)(a + 5)}$$ **Answer:** $$\boxed{\frac{-(a - 7)(a + 2)}{2(2a + 1)(a + 5)}}$$