Q. What is the sum of \( \frac{7x}{x^2 – 4} \) and \( \frac{2}{x + 2} \)?

We want to add the two rational expressions:

1. Write the expressions to be added:

   \(\frac{7x}{x^2 – 4}\) and \(\frac{2}{x + 2}\).

2. Factor any quadratic denominators to find a common denominator. Factor \(x^2 – 4\) as a difference of squares:

   \(x^2 – 4 = (x – 2)(x + 2)\).

3. Express both fractions with the common denominator \((x – 2)(x + 2)\). The first fraction already has that denominator:

   \(\frac{7x}{(x – 2)(x + 2)}\).

   The second fraction has denominator \(x + 2\), so multiply numerator and denominator by \((x – 2)\) to obtain the common denominator:

   \(\frac{2}{x + 2} = \frac{2(x – 2)}{(x + 2)(x – 2)}\).

4. Add the two fractions with the common denominator by adding their numerators:

   \(\frac{7x}{(x – 2)(x + 2)} + \frac{2(x – 2)}{(x – 2)(x + 2)} = \frac{7x + 2(x – 2)}{(x – 2)(x + 2)}\).

5. Expand and simplify the numerator:

   \(7x + 2(x – 2) = 7x + 2x – 4 = 9x – 4\).

   So the sum becomes

   \(\frac{9x – 4}{(x – 2)(x + 2)}\), which can be written as

   \(\frac{9x – 4}{x^2 – 4}\).

6. State the domain restrictions: the original expressions are undefined when any denominator is zero, so \(x\) cannot equal \(2\) or \(-2\).

Final simplified sum (with domain restriction \(x \neq \pm 2\)):

\(\frac{9x – 4}{x^2 – 4}\)