1. **State the problem:** Simplify the expression $$\frac{x^2 + 10x + 21}{2x^2 + 10x + 8} \div \frac{5x^2 - 20x - 105}{3x^2 + 15x + 12}$$ 2. **Rewrite division as multiplication by reciprocal:** $$\frac{x^2 + 10x + 21}{2x^2 + 10x + 8} \times \frac{3x^2 + 15x + 12}{5x^2 - 20x - 105}$$ 3. **Factor all polynomials:** - Numerator 1: $x^2 + 10x + 21 = (x + 3)(x + 7)$ - Denominator 1: $2x^2 + 10x + 8 = 2(x^2 + 5x + 4) = 2(x + 4)(x + 1)$ - Numerator 2: $3x^2 + 15x + 12 = 3(x^2 + 5x + 4) = 3(x + 4)(x + 1)$ - Denominator 2: $5x^2 - 20x - 105 = 5(x^2 - 4x - 21) = 5(x - 7)(x + 3)$ 4. **Substitute factored forms:** $$\frac{(x + 3)(x + 7)}{2(x + 4)(x + 1)} \times \frac{3(x + 4)(x + 1)}{5(x - 7)(x + 3)}$$ 5. **Multiply numerators and denominators:** $$\frac{(x + 3)(x + 7) \times 3(x + 4)(x + 1)}{2(x + 4)(x + 1) \times 5(x - 7)(x + 3)}$$ 6. **Cancel common factors:** - Cancel $(x + 3)$ numerator and denominator - Cancel $(x + 4)$ numerator and denominator - Cancel $(x + 1)$ numerator and denominator Intermediate step showing cancellation: $$\frac{\cancel{(x + 3)}(x + 7) \times 3\cancel{(x + 4)}\cancel{(x + 1)}}{2\cancel{(x + 4)}\cancel{(x + 1)} \times 5(x - 7)\cancel{(x + 3)}}$$ 7. **Simplify constants and remaining factors:** $$\frac{3(x + 7)}{2 \times 5 (x - 7)} = \frac{3(x + 7)}{10(x - 7)}$$ **Final answer:** $$\boxed{\frac{3(x + 7)}{10(x - 7)}}$$