1. **State the problem:** Simplify the expression $ \frac{6x + 27}{18x^2 + 36x} \div \frac{16x + 72}{2x + 4} $. 2. **Rewrite the division as multiplication by the reciprocal:** $$\frac{6x + 27}{18x^2 + 36x} \times \frac{2x + 4}{16x + 72}$$ 3. **Factor all polynomials:** - Numerator 1: $6x + 27 = 3(2x + 9)$ - Denominator 1: $18x^2 + 36x = 18x(x + 2)$ - Numerator 2: $2x + 4 = 2(x + 2)$ - Denominator 2: $16x + 72 = 8(2x + 9)$ 4. **Substitute factored forms:** $$\frac{3(2x + 9)}{18x(x + 2)} \times \frac{2(x + 2)}{8(2x + 9)}$$ 5. **Multiply the fractions:** $$\frac{3(2x + 9) \times 2(x + 2)}{18x(x + 2) \times 8(2x + 9)}$$ 6. **Cancel common factors:** - Cancel $2x + 9$ from numerator and denominator: $$\frac{3\cancel{(2x + 9)} \times 2(x + 2)}{18x(x + 2) \times 8\cancel{(2x + 9)}}$$ - Cancel $x + 2$ from numerator and denominator: $$\frac{3 \times 2\cancel{(x + 2)}}{18x\cancel{(x + 2)} \times 8}$$ 7. **Simplify constants:** $$\frac{3 \times 2}{18x \times 8} = \frac{6}{144x}$$ 8. **Simplify the fraction:** $$\frac{\cancel{6}^1}{\cancel{144}^{24}x} = \frac{1}{24x}$$ **Final answer:** $$\boxed{\frac{1}{24x}}$$