Is there a poly-time algorithm for expanding out polynomials
For the particular polynomial in the question, we have the multinomial theorem.
$$ (x_1+x_2+\cdots+x_n)^m = \sum_{k_i\geq 0,\, k_1+k_2+\cdots+k_n=m} \binom{m}{k_1,k_2,
\cdots,k_n} x_1^{k_1}x_2^{k_2}\cdots x_n^{k_n}$$
where the coefficients are knowns as multinomial coefficients, and can be computed by
$$ \binom{m}{k_1,k_2,\cdots,k_n} = \frac{m!}{k_1!\,k_2!\cdots k_n!}$$

For most of general products of multivariable polynomials, there is no such closed formula for their expansion.