\[\left( \frac{2x + 5}{x^{2} + 4x + 4} - \frac{x + 3}{x^{2} + 2x} \right)\ :\frac{x^{2} - 6}{x^{3} - 4x} = \frac{x - 2}{x + 2}\]
\[Преобразуем\ левую\ часть:\]
\[\frac{2x + 5}{x^{2} + 4x + 4} - \frac{x + 3}{x^{2} + 2x} =\]
\[= \frac{2x + 5^{\backslash x}}{(x + 2)^{2}} - \frac{x + 3^{\backslash x + 2}}{x(x + 2)} =\]
\[= \frac{2x^{2} + 5x - x^{2} - 3x - 2x - 6}{x(x + 2)^{2}} =\]
\[= \frac{x^{2} - 6}{x(x + 2)^{2}};\]
\[\frac{x^{2} - 6}{x(x + 2)^{2}}\ :\ \frac{x^{2} - 6}{x^{3} - 4x} =\]
\[= \frac{\left( x^{2} - 6 \right)x\left( x^{2} - 4 \right)}{x(x + 2)^{2}\left( x^{2} - 6 \right)} =\]
\[= \frac{(x - 2)(x + 2)}{(x + 2)^{2}} = \frac{x - 2}{x + 2}.\]
\[Тождество\ доказано.\ \]