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This is an instance of Watson's quintuple product identity (also Macdonald identity for $BC_1$): $$\prod_{n\geq 1}(1-s^n)(1-s^nt)(1-s^{n-1}t^{-1})(1-s^{2n-1}t^2)(1-s^{2n-1}t^{-2})=\sum_{n\in \mathbb Z}s^{\frac{3n^2+n}{2}}(t^{3n}-t^{-3n-1}).$$ By plugging in $t=x^{-1}$ and $s=-x^{4}$ this becomes exactly your identity.