Question 25717
When you had (a+b)(a-b)=b(a-b) you divided both sides
by (a-b) to get a+b=b.
But in doing that you divided by zero because you 
started the argument by saying a=b.
Dividing by zero is an illegal move, i.e. it is 
meaningless; so you ended up with a meaningless
conclusion.

Cheers,
Stan H.