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) \( \frac{x}{x-2}-\frac{1}{x+2}+\frac{8}{4-x^{2}} \)

Let's simplify each expression step by step. --- ### **a)** \[ \frac{x^{2} + 5x + 6}{x^{2} - 9} \div \frac{x^{2} - 4}{x^{2} - 5x + 6} \] **Step 1: Rewrite the Division as Multiplication** Dividing by a fraction is the same as multiplying by its reciprocal: \[ \frac{x^{2} + 5x + 6}{x^{2} - 9} \times \frac{x^{2} - 5x + 6}{x^{2} - 4} \] **Step 2: Factor All Quadratic Expressions** - \(x^{2} + 5x + 6 = (x + 2)(x + 3)\) - \(x^{2} - 9 = (x - 3)(x + 3)\) *(Difference of squares)* - \(x^{2} - 5x + 6 = (x - 2)(x - 3)\) - \(x^{2} - 4 = (x - 2)(x + 2)\) *(Difference of squares)* **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 Common Factors** - \(x + 3\) cancels out. - \(x - 3\) cancels out. - \(x + 2\) cancels out. - \(x - 2\) cancels out. **Final Simplified Form:** \[ 1 \] --- ### **b)** \[ \frac{x}{x - 2} - \frac{1}{x + 2} + \frac{8}{4 - x^{2}} \] **Step 1: Factor the Denominator Where Possible** Notice that \(4 - x^{2}\) is a difference of squares: \[ 4 - x^{2} = (2 - x)(2 + x) = -(x - 2)(x + 2) \] **Step 2: Identify the Common Denominator** The denominators are \(x - 2\), \(x + 2\), and \(4 - x^{2}\). The common denominator is \((x - 2)(x + 2)\). **Step 3: Rewrite Each Fraction with the Common Denominator** \[ \frac{x}{x - 2} = \frac{x(x + 2)}{(x - 2)(x + 2)} \] \[ -\frac{1}{x + 2} = -\frac{(x - 2)}{(x - 2)(x + 2)} \] \[ \frac{8}{4 - x^{2}} = \frac{8}{-(x - 2)(x + 2)} = -\frac{8}{(x - 2)(x + 2)} \] **Step 4: Combine the Fractions** \[ \frac{x(x + 2) - (x - 2) - 8}{(x - 2)(x + 2)} \] **Step 5: Simplify the Numerator** \[ x(x + 2) - (x - 2) - 8 = x^2 + 2x - x + 2 - 8 = x^2 + x - 6 \] **Step 6: Factor the Numerator** \[ x^2 + x - 6 = (x - 2)(x + 3) \] **Step 7: Substitute Back and Cancel Common Factors** \[ \frac{(x - 2)(x + 3)}{(x - 2)(x + 2)} = \frac{x + 3}{x + 2} \quad \text{(assuming } x \neq 2, -2\text{)} \] **Final Simplified Form:** \[ \frac{x + 3}{x + 2} \] --- **Summary of Simplified Expressions:** - **a)** \(1\) - **b)** \(\dfrac{x + 3}{x + 2}\)