x ≠ 2

x ≠ - 2

[tex] \frac{x}{x + 2} + \frac{x + 2}{x - 2} = \frac{8}{ {x}^{2} - 4} \\ \frac{x}{x + 2} + \frac{x + 2}{x - 2} - \frac{8}{(x - 2)(x + 2)} = 0 \\ \frac{x(x - 2) + (x + 2) {}^{2} - 8 }{(x - 2)(x + 2)} = 0 \\ \frac{ {x}^{2} - 2x + {x}^{2} + 4x + 4 - 8}{(x - 2)(x + 2)} = 0 \\ \frac{2 {x}^{2} + 2x - 4 }{(x - 2)(x + 2)} = 0 \\ 2 {x}^{2} + 2x - 4 = 0 \\ {x}^{2} + x - 2 = 0 \\ a = 1\\ b = 1\\ c = - 2 \\ D = {b}^{2} - 4ac = 1 {}^{2} - 4 \times 1 \times ( - 2) = 1 + 8 = 9 \\ x_{1} = \frac{ - 1 + 3}{2 \times 1} = \frac{2}{2} = 1 \\ x_{2} = \frac{ - 1 - 3}{2 \times 1} = - \frac{4}{2} = - 2[/tex]

Второй корень не подходит

Ответ: х = 1