1. **State the problem:** Simplify the expression $$\frac{10x}{x^2+3x} \div \frac{15x}{x^2 - x - 12}$$ as a single fraction in simplest form. 2. **Rewrite division as multiplication:** Dividing by a fraction is the same as multiplying by its reciprocal. $$\frac{10x}{x^2+3x} \times \frac{x^2 - x - 12}{15x}$$ 3. **Factor all polynomials:** - Factor denominator $x^2 + 3x = x(x+3)$ - Factor numerator $x^2 - x - 12 = (x-4)(x+3)$ So the expression becomes: $$\frac{10x}{x(x+3)} \times \frac{(x-4)(x+3)}{15x}$$ 4. **Multiply numerators and denominators:** $$\frac{10x \times (x-4)(x+3)}{x(x+3) \times 15x}$$ 5. **Cancel common factors:** - Cancel $x+3$ from numerator and denominator - Cancel $x$ from numerator and denominator Intermediate step showing cancellation: $$\frac{10\cancel{x} \times (x-4)\cancel{(x+3)}}{\cancel{x}\cancel{(x+3)} \times 15x} = \frac{10 (x-4)}{15 x}$$ 6. **Simplify coefficients:** $$\frac{\cancel{10}^{2} (x-4)}{\cancel{15}^{3} x} = \frac{2 (x-4)}{3 x}$$ 7. **Final answer:** $$\boxed{\frac{2(x-4)}{3x}}$$ This is the simplest form of the given expression.