add or subtract as indicate \( \frac{5 x}{x+2}+\frac{17}{x-2}-\frac{40}{x^{2}-4} \)

To simplify the expression \( \frac{5 x}{x+2}+\frac{17}{x-2}-\frac{40}{x^{2}-4} \), we first note that \( x^2 - 4 \) can be factored as \( (x-2)(x+2) \). Thus we rewrite the expression: \[ \frac{5x}{x+2} + \frac{17}{x-2} - \frac{40}{(x-2)(x+2)} \] Next, we need a common denominator which is \( (x-2)(x+2) \). We can rewrite each term with the common denominator: 1. The first term: \[ \frac{5x}{x+2} \cdot \frac{x-2}{x-2} = \frac{5x(x-2)}{(x-2)(x+2)} = \frac{5x^2 - 10x}{(x-2)(x+2)} \] 2. The second term: \[ \frac{17}{x-2} \cdot \frac{x+2}{x+2} = \frac{17(x+2)}{(x-2)(x+2)} = \frac{17x + 34}{(x-2)(x+2)} \] 3. The third term is already in the correct form: \[ -\frac{40}{(x-2)(x+2)} \] Now, we can combine all the terms: \[ \frac{5x^2 - 10x + 17x + 34 - 40}{(x-2)(x+2)} \] Now combine like terms in the numerator: \[ 5x^2 + 7x + (34 - 40) = 5x^2 + 7x - 6 \] Thus, the expression simplifies to: \[ \frac{5x^2 + 7x - 6}{(x-2)(x+2)} \] The final answer is: \[ \frac{5x^2 + 7x - 6}{(x-2)(x+2)} \]