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We consider the problem of quantifying the variance in the number of molecules of a species, in biochemical reactions with nonlinear reaction rates. We address this problem for a particular configuration where a species is formed with bursts, with a nonlinear rate that depends on another spontaneously formed species. By making use of an appropriately formulated expansion based on the Newton series, in conjunction with spectral properties of the master equation, we derive an analytical expression that provides a hard bound for the variance. We also show that this bound is exact when the propensities are linear. Furthermore, numerical simulations demonstrate that this is very close to the actual variance.