$$b(a+b)^m = \sum_{j=0}^m \binom{m}{j}a^{m-j}b^{j+1}= \sum_{k=1}^m \binom{m}{k-1}a^{m+1-k}b^{k}+b^{m+1}$$
I believe $j = k-1$ though the book does say that. This is related to proving the binomial theorem inductively. How does the $b^{m+1}$ at the end get there? I would expect that to not be there and $a^{m}b$ to be added before the summation. I must be misunderstanding something fundamental here.