(a) \( \frac{2 x^{3}-2 x}{2 x^{4}+4 x^{3}-2 x^{2}-4 x} \)

Let's simplify the expression step by step. First, factor out common terms in both the numerator and the denominator: Numerator: \(2x(x^{2}-1) = 2x(x-1)(x+1)\) Denominator: \(2x( x^{3} + 2x^{2} - x - 2) \) Now let's factor \(x^{3} + 2x^{2} - x - 2\) using grouping: Group it as: \((x^{3} + 2x^{2}) + (-x - 2)\) Factoring gives: \(x^{2}(x + 2) - 1(x + 2)\) This combines to: \((x + 2)(x^{2} - 1) = (x + 2)(x - 1)(x + 1)\) Now putting it all together, our expression becomes: \[ \frac{2x(x - 1)(x + 1)}{2x(x + 2)(x - 1)(x + 1)} \] Notice that \(2x\), \((x - 1)\), and \((x + 1)\) cancel out from the numerator and denominator: \[ \frac{1}{x + 2} \] Thus, the simplified expression is: \[ \frac{1}{x + 2} \quad \text{for} \quad x \neq -2, 0, 1, -1. \]