\[\boxed{\text{232\ (232).\ }\text{Еуроки\ -\ ДЗ\ без\ мороки}}\]
\[Преобразуем\ левую\ часть\ \]
\[равенства:\]
\[\frac{a^{2}x + a^{2}y + b^{2}y + b^{2}x}{\left( a^{2} - b^{2} \right)(x + y)} =\]
\[= \frac{a^{2} + b^{2}}{a^{2} - b^{2}}\]
\[\frac{a^{2}(x + y) + b^{2}(x + y)}{\left( a^{2} - b^{2} \right)(x + y)} =\]
\[= \frac{a^{2} + b^{2}}{a^{2} - b^{2}}\]
\[\frac{a^{2} + b^{2}}{a^{2} - b^{2}} = \frac{a^{2} + b^{2}}{a^{2} - b^{2}}\]
\[Что\ и\ требовалось\ доказать.\]