I have trouble with the following: $$P(t) = \sum_{i=0}^{n-r}c_i^r(tB^{n-r-1}_{i-1}(t)+(1-t)B_i^{n-r-1}(t))$$
$$= \sum_{i=0}^{n-r-1}(tc^r_{i+1}+(1-t)c_i^r)B_i^{n-r-1}(t),$$ where $c_i$ are constants, and $B_i^n(t)$ are the Bernstein polynomials.
I am confused on how the to reach the second equality from the first one. It seems like just a matter of changing the indexing (reducing the sum index from (n-r) to n-(r+1), but I can't decipher how exactly.
Hint: \begin{align*} \sum_{i=0}^{n-r}c_i^r&(tB^{n-r-1}_{i-1}(t) +(1-t)B_i^{n-r-1}(t)) \\ &= \sum_{i=0}^{n-r}c_i^rtB^{n-r-1}_{i-1}(t)+\sum_{i=0}^{n-r}c_i^r(1-t)B_i^{n-r-1}(t) \\ &= \sum_{i=-1}^{n-r-1}c_{i+1}^rtB^{n-r-1}_i(t)+\sum_{i=0}^{n-r}c_i^r(1-t)B_i^{n-r-1}(t) \\ &= c_0^rtB^{n-r-1}_{-1}(t)+\sum_{i=0}^{n-r-1}c_{i+1}^rtB^{n-r-1}_i(t)+\sum_{i=0}^{n-r-1}c_i^r(1-t)B_i^{n-r-1}(t)+c_{n-r}^r(1-t)B_{n-r}^{n-r-1}(t)\dots. \end{align*}