[tex] \frac{ {x}^{2} + x }{ {x}^{2} - x} = \frac{x(x + 1)}{x(x - 1)} = \frac{x + 1}{x - 1} \\ \frac{ {x}^{2} - x}{ {x}^{3} - x} = \frac{x(x - 1)}{x( {x}^{2} - 1)} = \frac{x - 1}{(x - 1)(x + 1)} = \frac{1}{x + 1} [/tex]
[tex] \frac{ {x}^{3} -x }{ {x}^{2} + x} = \frac{x( {x}^{2} - 1)}{x(x + 1)} = \frac{(x - 1)(x + 1)}{x + 1} = x - 1[/tex]