1. **State the problem:** Simplify the expression $$\frac{3x + 6}{8} \div \frac{5x + 10}{6}$$. 2. **Recall the rule for dividing fractions:** Dividing by a fraction is the same as multiplying by its reciprocal. So, $$\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c}$$. 3. **Apply the rule:** $$\frac{3x + 6}{8} \div \frac{5x + 10}{6} = \frac{3x + 6}{8} \times \frac{6}{5x + 10}$$. 4. **Factor expressions where possible:** $$3x + 6 = 3(x + 2)$$ $$5x + 10 = 5(x + 2)$$ 5. **Rewrite the expression with factored terms:** $$\frac{3(x + 2)}{8} \times \frac{6}{5(x + 2)}$$. 6. **Cancel common factors:** The term $(x + 2)$ appears in numerator and denominator, so it cancels out. $$\frac{3}{8} \times \frac{6}{5}$$. 7. **Multiply numerators and denominators:** $$\frac{3 \times 6}{8 \times 5} = \frac{18}{40}$$. 8. **Simplify the fraction:** Both numerator and denominator are divisible by 2. $$\frac{18 \div 2}{40 \div 2} = \frac{9}{20}$$. **Final answer:** $$\frac{9}{20}$$.