I want to show that $ \sum_{i=0}^n {n\choose i} \left(x^{i+1}y^{n-i} + x^i y^{n-i+1}\right) = \sum_{i=0}^{n+1}{n+1 \choose i}x^i y^{n+1-i}. $ I was thinking to split the sum on the LHR into 2 sums and then perform an index transformation on the first sum, so I would be able to use Pascal's rule. Unfortunately I wasn't able to find the right transformation so far, as I always end up with something like $\sum_{i=0}^{n+1}{n\choose i-1}$ for the first sum, which doesn't make sense for the binomial coefficient (because $i-1=-1$).
Any hints?