Disprove 0=1

Someone has claimed that they proved that 0=1! Of course, they must have made a mistake somewhere. Which step below contains their first mistake?

$a+b=1\\ { (a+b) }^{ 2 }={ 1 }\\ a{ (a+b) }+b{ (a+b) }={ 1 }\\ { a }^{ 2 }+{ b }^{ 2 }+2ab=1\\ { a }^{ 2 }+{ b }^{ 2 }+2ab=a+b\\ { a }^{ 2 }+{ b }^{ 2 }+2ab-(a+b)=a+b-(a+b)\\ { a }^{ 2 }+ab-a+{ b }^{ 2 }+ab-b=0\\ a(a+b-1)+b(b+a-1)=0\\ (a+b)(a+b-1)=0\\ a+b=0\\ a+b-1=0\\ 1=0$