1. **State the problem:** Multiply and divide the rational expressions: $$\frac{x-1}{x^2+5x+6} \cdot \frac{x^2+x-2}{3} \div \frac{x-1}{3x}$$ 2. **Rewrite the division as multiplication by the reciprocal:** $$\frac{x-1}{x^2+5x+6} \cdot \frac{x^2+x-2}{3} \cdot \frac{3x}{x-1}$$ 3. **Factor all polynomials where possible:** - Factor the quadratic in the denominator: $$x^2+5x+6 = (x+2)(x+3)$$ - Factor the quadratic in the numerator: $$x^2+x-2 = (x+2)(x-1)$$ 4. **Substitute the factored forms:** $$\frac{x-1}{(x+2)(x+3)} \cdot \frac{(x+2)(x-1)}{3} \cdot \frac{3x}{x-1}$$ 5. **Multiply all numerators and denominators:** Numerator: $$(x-1)(x+2)(x-1)(3x)$$ Denominator: $$3(x+2)(x+3)(x-1)$$ 6. **Write the full expression:** $$\frac{(x-1)(x+2)(x-1)(3x)}{3(x+2)(x+3)(x-1)}$$ 7. **Cancel common factors:** - Cancel one $(x-1)$ from numerator and denominator: $$\frac{\cancel{(x-1)}(x+2)(x-1)(3x)}{3(x+2)(x+3)\cancel{(x-1)}}$$ - Cancel $(x+2)$ from numerator and denominator: $$\frac{(x-1)\cancel{(x+2)}(3x)}{3\cancel{(x+2)}(x+3)}$$ 8. **Simplify the fraction:** $$\frac{(x-1)(3x)}{3(x+3)}$$ - Cancel 3 in numerator and denominator: $$\frac{(x-1)\cancel{3}x}{\cancel{3}(x+3)}$$ 9. **Final simplified expression:** $$\frac{x(x-1)}{x+3}$$ **Answer:** $$\boxed{\frac{x(x-1)}{x+3}}$$