Represent the following mixed infinite decimal periodic fractions as common fractions:
$\frac{2 b + a - \frac{4 a^{2} - b^{2}}{a}}{b^{3} + 2 a b^{2} - 3 a^{2} b} \cdot \frac{a^{3} b - 2 a^{2} b^{2} + a b^{3}}{a^{2} - b^{2}}$

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Solution

$\frac{2 b + a - \frac{4 a^{2} - b^{2}}{a}}{b^{3} + 2 a b^{2} - 3 a^{2} b} \cdot \frac{a^{3} b - 2 a^{2} b^{2} + a b^{3}}{a^{2} - b^{2}}$
$= \frac{\frac{2 b + a^{2} - 4 a^{2} + b^{2}}{a}}{b (b^{2} + 2 a b - 3 a^{2})} \cdot \frac{a b (a^{2} - 2 a b + b^{2})}{a^{2} - b^{2}}$
$= \frac{2 a b + b^{2} - 3 a^{2}}{a b (b^{2} + 2 a b - 3 a^{2})} \cdot \frac{a b (a - b)^{2}}{a^{2} - b^{2}}$
$= \frac{(a - b)^{2}}{(a + b) (a - b)}$
$= \frac{a - b}{a + b}$

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