$\frac{x + 5}{x^2 - 5x} - \frac{x - 5}{2x^2 + 10x} = \frac{x + 25}{2x^2 - 50}$
$\frac{x + 5}{x^2 - 5x} - \frac{x - 5}{2x^2 + 10x} - \frac{x + 25}{2x^2 - 50} = 0$
$\frac{x + 5}{x(x - 5)} - \frac{x - 5}{2x(x + 5)} - \frac{x + 25}{2(x^2 - 25)} = 0$
$\frac{x + 5}{x(x - 5)} - \frac{x - 5}{2x(x + 5)} - \frac{x + 25}{2(x - 5)(x + 5)} = 0$
$\frac{2(x + 5)^2 - (x - 5)^2 - x(x + 25)}{2x(x - 5)(x + 5)} = 0$
$\frac{2(x^2 + 10x + 25) - (x^2 - 10x + 25) - x^2 - 25x}{2x(x - 5)(x + 5)} = 0$
$\frac{2x^2 + 20x + 50 - x^2 + 10x - 25 - x^2 - 25x}{2x(x - 5)(x + 5)} = 0$
$\frac{5x + 25}{2x(x - 5)(x + 5)} = 0$
$\frac{5(x + 5)}{2x(x - 5)(x + 5)} = 0$
$\frac{5}{2x(x - 5)} = 0$
5 ≠ 0
Ответ: нет корней

$\frac{2}{x^2 - 9} - \frac{1}{2x^2 - 12x + 18} = \frac{3}{2x^2 + 6x}$
$\frac{2}{x^2 - 9} - \frac{1}{2x^2 - 12x + 18} - \frac{3}{2x^2 + 6x} = 0$
$\frac{2}{(x - 3)(x + 3)} - \frac{1}{2(x^2 - 6x + 9)} - \frac{3}{2x(x + 3)} = 0$
$\frac{2}{(x - 3)(x + 3)} - \frac{1}{2(x - 3)^2} - \frac{3}{2x(x + 3)} = 0$
$\frac{2 * 2x(x - 3) - x(x + 3) - 3(x - 3)^2}{2x(x - 3)^2(x + 3)} = 0$
$\frac{4x^2 - 12x - x^2 - 3x - 3(x^2 - 6x + 9)}{2x(x - 3)^2(x + 3)} = 0$
$\frac{3x^2 - 15x - 3x^2 + 18x - 27}{2x(x - 3)^2(x + 3)} = 0$
$\frac{3x - 27}{2x(x - 3)^2(x + 3)} = 0$
$\begin{equation*} \begin{cases} 2x ≠ 0 &\\ x - 3 ≠ 0 &\\ x + 3 ≠ 0 &\\ 3x - 27 = 0 & \end{cases} \end{equation*}$
$\begin{equation*} \begin{cases} x ≠ 0 &\\ x ≠ 3 &\\ x ≠ -3 &\\ 3x = 27 & \end{cases} \end{equation*}$
$\begin{equation*} \begin{cases} x ≠ 0 &\\ x ≠ 3 &\\ x ≠ -3 &\\ x = 9 & \end{cases} \end{equation*}$
Ответ: x = 9

$\frac{9x + 12}{x^3 - 64} - \frac{1}{x - 4} = \frac{1}{x^2 + 4x + 16}$
$\frac{9x + 12}{(x - 4)(x^2 + 4x + 16)} - \frac{1}{x - 4} - \frac{1}{x^2 + 4x + 16} = 0$
$\frac{9x + 12 - (x^2 + 4x + 16) - (x - 4)}{(x - 4)(x^2 + 4x + 16)} = 0$
$\frac{9x + 12 - x^2 - 4x - 16 - x + 4}{(x - 4)(x^2 + 4x + 16)} = 0$
$\frac{4x - x^2}{(x - 4)(x^2 + 4x + 16)} = 0$
$\frac{x(4 - x)}{(x - 4)(x^2 + 4x + 16)} = 0$
$\frac{-x(x - 4)}{(x - 4)(x^2 + 4x + 16)} = 0$
$\frac{-x}{x^2 + 4x + 16} = 0$
$\begin{equation*} \begin{cases} x^2 + 4x + 16 ≠ 0 &\\ -x = 0 & \end{cases} \end{equation*}$
$\begin{equation*} \begin{cases} x^2 + 4x + 16 ≠ 0 &\\ x = 0 & \end{cases} \end{equation*}$
Ответ: x = 0