Simplify the following expression as much as possible:

$$
\frac{\frac{1}{a-b}-\frac{2 a b}{a^{3}-a^{2} b+a b^{2}-b^{3}}}{\frac{a^{2}+a b}{a^{3}+a^{2} b+a b^{2}+b^{3}}+\frac{b}{a^{2}+b^{2}}}
$$
Our expression can also be written as

$$
\frac{\frac{1}{a-b}-\frac{2 a b}{(a-b)\left(a^{2}+b^{2}\right)}}{\frac{a(a+b)}{(a+b)\left(a^{2}+b^{2}\right)}+\frac{b}{a^{2}+b^{2}}}
$$

By extending the fraction with $(a+b)(a-b)\left(a^{2}+b^{2}\right)$

$$
\begin{gathered}
\frac{(a+b)\left(a^{2}+b^{2}\right)-2 a b(a+b)}{a\left(a^{2}-b^{2}\right)+b\left(a^{2}-b^{2}\right)}=\frac{(a+b)(a-b)^{2}}{\left(a^{2}-b^{2}\right)(a+b)}= \\
=\frac{(a-b)^{2}}{(a+b)(a-b)}=\frac{a-b}{a+b}
\end{gathered}
$$

András Bakos (Bp., VIII., Piarist g. II. o. t.)