You might consider the "sum-of-squares" approach.  The idea is to find a set of polynomials so that your expression is the sum of squares of the elements in the region of interest.  For your case, you could replace each $x_i$ with a new variable $z_i^2$; you are now asking if the corresponding 4-th degree unconstrained polynomial is non-negative.

This restatement may not sound like an improvement, but it turns out that SOS problems can be attacked using semidefinite programming techniques (e.g., see [this page][1]).  You can use a freely available SDP solvers.

This is a sufficient approach, i.e., it may prove that your original quadratic form is positive, but it can't disprove it.  Since you're trying to solve a specific problem, though, it may be worth the gamble.


  [1]: https://en.wikipedia.org/wiki/Sum-of-squares_optimization