3. Multiply or divide each. List any restrictions on the variables. a. \( \frac{2 x^{2}-32}{x^{2}-10 x+24} \cdot \frac{x^{2}-4 x-12}{10 x+20} \)

To solve the expression \[ \frac{2 x^{2}-32}{x^{2}-10 x+24} \cdot \frac{x^{2}-4 x-12}{10 x+20}, \] we will follow these steps: 1. **Factor each polynomial in the expression.** 2. **Identify any restrictions on the variables.** 3. **Multiply the fractions.** ### Step 1: Factor each polynomial **a. Factor \(2x^2 - 32\)** This can be factored as: \[ 2(x^2 - 16) = 2(x - 4)(x + 4). \] **b. Factor \(x^2 - 10x + 24\)** To factor this quadratic, we look for two numbers that multiply to \(24\) and add to \(-10\). The factors are \(-6\) and \(-4\): \[ x^2 - 10x + 24 = (x - 6)(x - 4). \] **c. Factor \(x^2 - 4x - 12\)** We need two numbers that multiply to \(-12\) and add to \(-4\). The factors are \(-6\) and \(2\): \[ x^2 - 4x - 12 = (x - 6)(x + 2). \] **d. Factor \(10x + 20\)** This can be factored as: \[ 10(x + 2). \] ### Step 2: Rewrite the expression with factored forms Now we can rewrite the original expression using the factored forms: \[ \frac{2(x - 4)(x + 4)}{(x - 6)(x - 4)} \cdot \frac{(x - 6)(x + 2)}{10(x + 2)}. \] ### Step 3: Simplify the expression Now we can multiply the fractions: \[ \frac{2(x - 4)(x + 4)(x - 6)(x + 2)}{(x - 6)(x - 4)10(x + 2)}. \] Next, we can cancel out the common factors: - \(x - 4\) in the numerator and denominator. - \(x - 6\) in the numerator and denominator. - \(x + 2\) in the numerator and denominator. After canceling, we have: \[ \frac{2(x + 4)}{10}. \] This simplifies to: \[ \frac{x + 4}{5}. \] ### Step 4: Identify restrictions on the variables The restrictions come from the denominators in the original expression. We need to ensure that the denominators do not equal zero: 1. From \(x^2 - 10x + 24 = 0\): - \(x - 6 = 0 \Rightarrow x \neq 6\) - \(x - 4 = 0 \Rightarrow x \neq 4\) 2. From \(10x + 20 = 0\): - \(x + 2 = 0 \Rightarrow x \neq -2\) ### Final Result The simplified expression is: \[ \frac{x + 4}{5} \] with restrictions: \[ x \neq 6, \quad x \neq 4, \quad x \neq -2. \]