1. **State the problem:** Simplify the expression $$\frac{x^2 - 7x + 12}{x^2 - 3x} \div \frac{x - 4}{x + 1}$$. 2. **Rewrite division as multiplication:** Dividing by a fraction is the same as multiplying by its reciprocal. $$\frac{x^2 - 7x + 12}{x^2 - 3x} \times \frac{x + 1}{x - 4}$$ 3. **Factor all polynomials:** - Numerator of first fraction: $$x^2 - 7x + 12 = (x - 3)(x - 4)$$ - Denominator of first fraction: $$x^2 - 3x = x(x - 3)$$ 4. **Substitute factored forms:** $$\frac{(x - 3)(x - 4)}{x(x - 3)} \times \frac{x + 1}{x - 4}$$ 5. **Cancel common factors:** - Cancel $(x - 3)$ in numerator and denominator. - Cancel $(x - 4)$ in numerator and denominator. Intermediate step showing cancellation: $$\frac{\cancel{(x - 3)}\cancel{(x - 4)}}{x\cancel{(x - 3)}} \times \frac{x + 1}{\cancel{x - 4}} = \frac{1}{x} \times (x + 1)$$ 6. **Multiply remaining factors:** $$\frac{1}{x} \times (x + 1) = \frac{x + 1}{x}$$ 7. **Final answer:** $$\boxed{\frac{x + 1}{x}}$$ **Note:** The domain excludes values that make any denominator zero: $x \neq 0, 3, 4, -1$.