Going from (1) to (2) isn't an obvious step, but it's true nonetheless:

(a-b)(a-b)'+(b-c)(b-c)'

(a-b)(a'-b')+(b-c)(b'-c')

aa'-ba'-ab'+bb'+bb'-cb'-bc'+cc'

Now add some extra terms:

aa'-ba'-ab'+bb'+bb'-cb'-bc'+cc'+ca'-ca'+ac'-ac'

And move everything around a bit:

aa'-ac'-ca'+cc'-ba'+bb'+ca'-cb'-ab'+ac'+bb'-bc'

a(a'-c')-c(a'-c')-b(a'-b')+c(a'-b')-a(b'-c')+b(b'-c')

(a-c)(a'-c')+(-b+c)(a'-b')+(-a+b)(b'-c')

(a-c)(a-c)'-(b-c)(a-b)'-(a-b)(b-c)'

Lastly, just add some more extra terms:

(a-b+b-c)(a-b+b-c)'-(b-c)(a-b)'-(a-b)(b-c)'

And that's it.