Activity 10 Simplify the following \( \frac{\text { expressions }}{\text { a) } \frac{x^{2}+5 x+6}{x^{2}-9} \div \frac{x^{2}-4}{x^{2}-5 x+6}} \) b) \( \frac{x}{r-2}-\frac{1}{x+2}+\frac{8}{4-x^{2}} \)

Let's simplify each part of **Activity 10** step by step. --- ### **Problem a)** Simplify: \[ \frac{x^{2}+5x+6}{x^{2}-9} \div \frac{x^{2}-4}{x^{2}-5x+6} \] **Step 1: Rewrite the Division as Multiplication** Dividing by a fraction is equivalent to multiplying by its reciprocal: \[ \frac{x^{2}+5x+6}{x^{2}-9} \times \frac{x^{2}-5x+6}{x^{2}-4} \] **Step 2: Factor Each Polynomial** - \( x^{2} + 5x + 6 = (x + 2)(x + 3) \) - \( x^{2} - 9 = (x - 3)(x + 3) \) - \( x^{2} - 5x + 6 = (x - 2)(x - 3) \) - \( x^{2} - 4 = (x - 2)(x + 2) \) **Step 3: Substitute the Factored Forms** \[ \frac{(x + 2)(x + 3)}{(x - 3)(x + 3)} \times \frac{(x - 2)(x - 3)}{(x - 2)(x + 2)} \] **Step 4: Cancel Out Common Factors** - \( (x + 3) \) cancels out. - \( (x - 3) \) cancels out. - \( (x - 2) \) cancels out. - \( (x + 2) \) cancels out. **Step 5: Simplify Remaining Terms** After cancellation, we're left with: \[ 1 \] **Final Answer for Part a):** \[ 1 \] --- ### **Problem b)** Simplify: \[ \frac{x}{x - 2} - \frac{1}{x + 2} + \frac{8}{4 - x^{2}} \] **Step 1: Factor the Denominators** - \( 4 - x^{2} = (2 - x)(2 + x) = -(x - 2)(x + 2) \) **Step 2: Rewrite the Expression with a Common Denominator** The common denominator is \( (x - 2)(x + 2) \). \[ \frac{x}{x - 2} = \frac{x(x + 2)}{(x - 2)(x + 2)} \] \[ -\frac{1}{x + 2} = -\frac{(x - 2)}{(x - 2)(x + 2)} \] \[ \frac{8}{4 - x^{2}} = -\frac{8}{(x - 2)(x + 2)} \] **Step 3: Combine the Fractions** \[ \frac{x(x + 2) - (x - 2) - 8}{(x - 2)(x + 2)} \] **Step 4: Simplify the Numerator** Expand and combine like terms: \[ x(x + 2) = x^2 + 2x \] \[ - (x - 2) = -x + 2 \] \[ x^2 + 2x - x + 2 - 8 = x^2 + x - 6 \] **Step 5: Factor the Numerator** \[ x^2 + x - 6 = (x + 3)(x - 2) \] **Step 6: Cancel Out Common Factors** \[ \frac{(x + 3)(x - 2)}{(x - 2)(x + 2)} = \frac{x + 3}{x + 2} \] **Final Answer for Part b):** \[ \frac{x + 3}{x + 2} \] --- ### **Summary of Results** - **a)** The simplified expression is: \[ 1 \] - **b)** The simplified expression is: \[ \frac{x + 3}{x + 2} \]