1. **Problem statement:** Multiply and simplify the expression $\frac{x^2 + 5x + 6}{4x - 12} \times \frac{x - 3}{x^2 + 2x - 3}$. 2. **Recall the formula and rules:** When multiplying fractions, multiply numerators together and denominators together: $$\frac{a}{b} \times \frac{c}{d} = \frac{a \times c}{b \times d}$$ Also, factor polynomials to simplify by canceling common factors. 3. **Factor each polynomial:** - Numerator 1: $x^2 + 5x + 6 = (x + 2)(x + 3)$ - Denominator 1: $4x - 12 = 4(x - 3)$ - Numerator 2: $x - 3$ (already factored) - Denominator 2: $x^2 + 2x - 3 = (x + 3)(x - 1)$ 4. **Rewrite the expression with factors:** $$\frac{(x + 2)(x + 3)}{4(x - 3)} \times \frac{x - 3}{(x + 3)(x - 1)}$$ 5. **Multiply numerators and denominators:** $$\frac{(x + 2)(x + 3)(x - 3)}{4(x - 3)(x + 3)(x - 1)}$$ 6. **Cancel common factors:** - Cancel $x - 3$ in numerator and denominator - Cancel $x + 3$ in numerator and denominator Intermediate step showing cancellation: $$\frac{(x + 2)\cancel{(x + 3)}\cancel{(x - 3)}}{4\cancel{(x - 3)}\cancel{(x + 3)}(x - 1)} = \frac{x + 2}{4(x - 1)}$$ 7. **Final simplified expression:** $$\frac{x + 2}{4(x - 1)}$$ This is the simplified product of the given rational expressions.