Докажите тождество: (a^2/(a+5)-a^3/(a^2+10a+25):(a/(a+5)-a^2/(a^2-25))=(5a-a^2)/(a+5).

\[\left( \frac{a^{2}}{a + 5} - \frac{a^{3}}{a^{2} + 10a + 25} \right)\ :\left( \frac{a}{a + 5} - \frac{a^{2}}{a^{2} - 25} \right) = \frac{5a - a^{2}}{a + 5}\]

\[Преобразуем\ левую\ часть:\]

\[\frac{a^{2}}{a + 5} - \frac{a^{3}}{a^{2} + 10a + 25} =\]

\[= \frac{{a^{2}}^{\backslash a + 5}}{a + 5} - \frac{a^{3}}{(a + 5)^{2}} =\]

\[= \frac{a^{2} + 5a^{2} - a^{3}}{(a + 5)^{2}} = \frac{5a^{2}}{(a + 5)^{2}};\]

\[\frac{a}{a + 5} - \frac{a^{2}}{a^{2} - 25} =\]

\[= \frac{a^{\backslash a - 5}}{a + 5} - \frac{a^{2}}{(a + 5)(a - 5)} =\]

\[= \frac{a^{2} - 5a - a^{2}}{a^{2} - 25} = - \frac{5a}{a^{2} - 25};\]

\[\frac{5a^{2}}{(a + 5)^{2}}\ :\left( - \frac{5a}{a^{2} - 25} \right) =\]

\[= - \frac{5a^{2}}{(a + 5)^{2}} \cdot \frac{a^{2} - 25}{5a} =\]

\[= - \frac{a(a - 5)(a + 5)}{(a + 5)^{2}} =\]

\[= \frac{- a(a - 5)}{a + 5} = \frac{5a - a^{2}}{a + 5}.\]

\[Что\ и\ требовалось\ доказать.\]