$\begin{align*}
p_i(x) &:= \sum_{j=1}^n (a_{i-1} - b_{j-1}) x^{i+j} = \sum_{j=0}^{n-1} (a_{i-1} - b_j) x^{i+j+1} \\
&= x^{i+1} \sum_{j=0}^{n-1} (a_{i-1} - b_j) x^j.
\end{align*}