You can rewrite your polynomial in squared variables (so it will be of degree 8) and homogenise it, obtaining a ternary form of degree 8.
There is a result of Hilbert in Acta Math. 17(1893) that applies here; he shows that a real ternary form of degree $2d$ satisfies $f(x,y,z)\geq 0$ for all $x,y,z$ iff it is representable as the rational function $p(x,y,z)/q(x,y,z)$, with $p$ and $q$ homogeneous globally nonnegative of degrees $4d-4$, resp., $2d-4$, and $p$ being a sum of squares of forms. (the result also says how many terms there are, etc)
Here, as $d=4$, we can assume that $q$ is a sum of squares of forms, as it is of degree 4 (by much more famous Hilbert's result from 1888).
We showed that this can be made into an algorithm, cf. http://www.optimization-online.org/DB_HTML/2001/11/399.html (published here: http://www.sciencedirect.com/science/article/pii/S0377221703005162)