1. **State the problem:** Simplify the product of the functions $f(x)$ and $g(x)$ given by $$ \frac{(x - 4)(x + 1)}{(x - 2)(x + 1)} \cdot \frac{(x - 3)(x - 2)}{(x - 4)(x + 2)} $$ and find the restricted values where the expressions are undefined. 2. **Recall the rules:** - You can multiply fractions by multiplying numerators and denominators. - Cancel common factors in numerator and denominator. - Restricted values are values of $x$ that make any denominator zero. 3. **Multiply the fractions:** $$ \frac{(x - 4)(x + 1)}{(x - 2)(x + 1)} \cdot \frac{(x - 3)(x - 2)}{(x - 4)(x + 2)} = \frac{(x - 4)(x + 1)(x - 3)(x - 2)}{(x - 2)(x + 1)(x - 4)(x + 2)} $$ 4. **Cancel common factors:** - $(x + 1)$ appears in numerator and denominator. - $(x - 4)$ appears in numerator and denominator. - $(x - 2)$ appears in numerator and denominator. Show cancellation step: $$ \frac{\cancel{(x - 4)}\cancel{(x + 1)}(x - 3)\cancel{(x - 2)}}{\cancel{(x - 2)}\cancel{(x + 1)}\cancel{(x - 4)}(x + 2)} = \frac{(x - 3)}{(x + 2)} $$ 5. **Simplified expression:** $$ \boxed{\frac{(x - 3)}{(x + 2)}} $$ 6. **Find restricted values:** - From original denominators: - $x - 2 = 0 \Rightarrow x = 2$ - $x + 1 = 0 \Rightarrow x = -1$ - $x - 4 = 0 \Rightarrow x = 4$ - $x + 2 = 0 \Rightarrow x = -2$ So, the restricted values are $x = -2, -1, 2, 4$. **Final answer:** $$ f(x) \cdot g(x) = \frac{(x - 3)}{(x + 2)}, \quad x \neq -2, -1, 2, 4 $$