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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+b1−a+ba−b2=(a−b)a+ba+b+b2a+b=a+b(a−b)(a+b)+b2=a+ba2−b2+b2=a2a+b=1−(a+b)2(a−b)2=(a+b)2(a+b)2−(a+b)2(a−b)2=(a+b)2(a+b)2−(a−b)2=(a+b)2(a2+2ab+b2)−(a2−2ab+b2)=4ab(a+b)2.

The whole fraction is therefore

a−b+b2a+b1−a+ba−b2=a2a+b4ab(a+b)2=a2a+b4ab(a+b)2=4ba(a+b).