There is an old theorem by George Pólya which says the following:
Theorem: If $p \in \mathbb R[x_1,\dots,x_n]$ is a homogenous polynomial which is positive on the positive octant, then for large $k$, the coefficients of the polynomial $(x_1 + \cdots +x_n)^k \cdot p(x_1,\dots,x_n)$ are positive.
Note that the condition in the theorem is necessary and sufficient since positivity of the coefficients of $(x_1 + \cdots +x_n)^k \cdot p(x_1,\dots,x_n)$ also implies that $p$ was positive on the positive octant.
For a reference or even a proof of this theorem you can look here. These are slides for a talk by Mari Castle called "Everything you’ve ever wanted to know about Pólya’s Theorem (but were afraid to ask)."