What is the common denominator of \( \frac{5}{x^{2}-4}-\frac{2}{x+2} \) in the complex fraction \( \frac{\frac{2}{x-2}-\frac{3}{x^{2}-4}}{\frac{5}{x^{2}-4}-\frac{2}{x+2}} \) ?

To find the common denominator of the expression \( \frac{5}{x^{2}-4}-\frac{2}{x+2} \), we first analyze the denominators involved. 1. The first fraction has a denominator of \( x^2 - 4 \), which can be factored as \( (x - 2)(x + 2) \). 2. The second fraction has a denominator of \( x + 2 \). To find a common denominator for \( \frac{5}{x^{2}-4} \) and \( \frac{2}{x+2} \), we need to take the least common multiple (LCM) of the denominators \( (x - 2)(x + 2) \) and \( x + 2 \). The LCM is determined as follows: - The factor \( (x - 2) \) appears in the first denominator but not in the second. - The factor \( (x + 2) \) appears in both denominators, but since we only take it once for the LCM, we count it once. Thus, the common denominator will be: \[ (x - 2)(x + 2) \] Putting it all together, the common denominator of \( \frac{5}{x^2-4}-\frac{2}{x+2} \) is \( (x - 2)(x + 2) \).