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From Förberedande kurs i matematik 1

First, we simplify the numerator and denominator for the whole fraction by rewriting as

a−b+b2a+b=a−ba+ba+b+b2a+b=a+ba−ba+b+b2a+ba2−b2+b2=a2a+b

1−a+ba−b2=1−a+b2a−b2=a+b2a+b2−a+b2a−b2=a+b2a+b2−a−b2=a+b2a2+2ab+b2−a2−2ab+b2=4aba+b2

The whole fraction is therefore

1−a+ba−b2a−b+b2a+b=a2a+b4aba+b2=a2a+b4aba+b2=4baa+b