General formula for $\prod (x+a_i)$

Question

What could be a general formula for this in terms of $x$ and $a_1,\; a_2\; \ldots\; a_{n-1}, \;a_n$? $$\prod_{i=0}^n(x+a_i)$$

I've tried solving it, but I'm lost at the sum-of-product-of-all-possible-combinations stage.

Answer 1

$$\prod_{i=1}^n(x+a_i)=\sum_{B\subseteq\{1,2,...,n\}}x^{n-|B|}\prod_{b\in B}a_b$$

Answer 2

I don't know if that is, what you want, but: One can see that the coefficient of $x^k$ in the expanded form is the sum of all $k$-term sums of $a_{i_\mu}$s. As a particular $i$ can appear only once among the $i_\mu$, we can order the $i_\mu$ strictly increasing, giving $$\prod_i (x+a_i) = \sum_{k=0}^n \left(\sum_{i_1 < \ldots < i_k} \prod_{\mu=1}^k a_{i_\mu}\right) x^k $$