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

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To find the common denominator of the expression $$ \frac{5}{x^{2}-4}-\frac{2}{x+2} $$, we first need to identify the denominators involved in this expression.

1. The first term has a denominator of $$ x^2 - 4 $$.
2. The second term has a denominator of $$ x + 2 $$.

Next, we factor $$ x^2 - 4 $$:
$$
x^2 - 4 = (x - 2)(x + 2)
$$

Now, we have the denominators:
- $$ x^2 - 4 = (x - 2)(x + 2) $$
- $$ x + 2 $$

To find the common denominator, we need to take the least common multiple (LCM) of these denominators. The LCM will include each factor at its highest power.

The factors are:
- From $$ x^2 - 4 $$: $$ (x - 2) $$ and $$ (x + 2) $$
- From $$ x + 2 $$: $$ (x + 2) $$

Thus, the common denominator is:
$$
(x - 2)(x + 2)
$$

Now, we can summarize the common denominator of $$ \frac{5}{x^{2}-4}-\frac{2}{x+2} $$ as:
$$
\text{Common Denominator} = (x - 2)(x + 2)
$$
The common denominator is $$ (x - 2)(x + 2) $$.

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

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