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)."

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)."