We give a combinatorial proof that\(\left[ {\begin{array}{*{20}c} a \\ k \\ \end{array} } \right]_q \left[ {\begin{array}{*{20}c} b \\ l \\ \end{array} } \right]_q - \left[ {\begin{array}{*{20}c} a \\ {k - 1} \\ \end{array} } \right]_q \left[ {\begin{array}{*{20}c} b \\ {l + 1} \\ \end{array} } \right]_q \) is a polynomial inq with nonnegative coefficients for nonnegative integersa, b, k, l witha≥b andl≥k. In particular, fora=b=n andl=k, this implies theq-log-concavity of the Gaussian binomial coefficients\(\left[ {\begin{array}{*{20}c} n \\ k \\ \end{array} } \right]_q \), which was conjectured byButler (Proc. Amer. Math. Soc. 101 (1987), 771–775).