The long-standing Gaussian product inequality (GPI) conjecture states that E[ n j=1 X] for any centered Gaussian random vector (X 1 , . . . , X n ) and m 1 , . . . , m n ∈ N. In this note, we describe a computational algorithm involving sums-of-squares representations of multivariate polynomials that can be used to resolve the GPI conjecture. To exhibit the power of this novel method, we apply it to prove two special GPIs: E