1. **State the problem:** We are given the function $$f(x) = 2x + \frac{x}{x^{2} - 1}$$ and we want to understand its behavior or simplify it if possible. 2. **Recall the formula and rules:** The function is a sum of a linear term and a rational function. The denominator $$x^{2} - 1$$ can be factored using the difference of squares formula: $$x^{2} - 1 = (x - 1)(x + 1)$$ 3. **Rewrite the function:** $$f(x) = 2x + \frac{x}{(x - 1)(x + 1)}$$ 4. **Find a common denominator to combine terms:** Rewrite $$2x$$ as $$\frac{2x(x - 1)(x + 1)}{(x - 1)(x + 1)}$$ to have the same denominator: $$f(x) = \frac{2x(x - 1)(x + 1)}{(x - 1)(x + 1)} + \frac{x}{(x - 1)(x + 1)}$$ 5. **Expand the numerator of the first fraction:** $$(x - 1)(x + 1) = x^{2} - 1$$ So, $$2x(x^{2} - 1) = 2x^{3} - 2x$$ 6. **Combine the fractions:** $$f(x) = \frac{2x^{3} - 2x + x}{(x - 1)(x + 1)} = \frac{2x^{3} - x}{(x - 1)(x + 1)}$$ 7. **Factor the numerator:** $$2x^{3} - x = x(2x^{2} - 1)$$ 8. **Final simplified form:** $$f(x) = \frac{x(2x^{2} - 1)}{(x - 1)(x + 1)}$$ 9. **Domain restrictions:** The function is undefined at $$x = 1$$ and $$x = -1$$ because the denominator is zero there. **Answer:** $$f(x) = \frac{x(2x^{2} - 1)}{(x - 1)(x + 1)}$$ with $$x \neq \pm 1$$.