I had that a(b+c) = ab + ac. This is a common postulate in some books. I take it for granted most of the time. Surely that's not what's being proved here. What we're trying to prove is that:
a*0 = 0 for all a.
Using b=c=0 in my second line of manipulations, a*(0+0)= a*0 + a*0.
This is close to what your friend had, except that there were two steps missing. My approach is perfectly valid and in fact is also shown in some textbooks.