Verified answer

[tex] \frac{a - 2}{ {a}^{2} - 4} - \frac{a - 2}{4 - {a}^{2} } = \frac{a - 2}{(a - 2)(a + 2)} - \frac{a - 2}{ - (a - 2)(2 + a)} = \frac{1}{a + 2} - \frac{ - 1}{2 + a} = \frac{2}{a + 2} [/tex]

[tex] \frac{15x - 2}{10 {x}^{2} } + \frac{5 + x}{5 {x}^{3} } = \frac{x(15x - 2) + 2(5 + x)}{10 {x}^{3} } = \frac{15 {x}^{2} - 2x + 10 + 2x }{10 {x}^{3} } = \frac{15 {x}^{2} + 10}{10 {x}^{3} } = \frac{5(3 {x}^{2} + 2)}{10 {x}^{3} } = \frac{3 {x}^{2} + 2}{2 {x}^{3} } [/tex]

[tex] \frac{7}{ {x}^{2} + x} + \frac{13}{x + 1} = \frac{7 + 13x}{ {x}^{2} + x } [/tex]