15. \( \frac{5 x}{x+1}-\frac{2 x}{x^{2}-1} \)

To tackle the expression \( \frac{5 x}{x+1}-\frac{2 x}{x^{2}-1} \), first notice that \( x^{2}-1 \) can be factored into \( (x-1)(x+1) \). To simplify the subtraction, we need a common denominator, which will be \( (x+1)(x-1) \). Now, rewrite each fraction with this common denominator: 1. The first fraction becomes: \[ \frac{5x}{x+1} \cdot \frac{x-1}{x-1} = \frac{5x(x-1)}{(x+1)(x-1)} \] 2. The second fraction becomes: \[ \frac{2x}{(x-1)(x+1)} \text{ (already has the common denominator)} \] Now, combine the two fractions: \[ \frac{5x(x-1) - 2x}{(x+1)(x-1)} \] Simplifying the numerator: \[ 5x^2 - 5x - 2x = 5x^2 - 7x \] So the expression reduces to: \[ \frac{5x^2 - 7x}{(x+1)(x-1)} \] Lastly, you can factor out \( x \) in the numerator: \[ \frac{x(5x - 7)}{(x+1)(x-1)} \] And there you have it! The fully simplified expression is: \[ \frac{x(5x - 7)}{(x+1)(x-1)} \]