This paper is concerned with the oscillatory behavior of all solutions of the second-order mixed nonlinear neutral dynamic equation with damping $(r(t)\phi (z^\Delta (t)))^\Delta + p(t)\phi (z^\Delta (t)) + f (t,x(\tau_1(t))) + g(t,x(\tau_2(t))) = 0$, where $\phi (s) = |s|^{\gamma-1}$ and $z(t) = x(t) + p_1(t)x(\eta_1(t)) + p_2(t)x(\eta_2(t))$. Our results complement and improve the results reported in [[1],[2], [6]-[9], [12], [13], [15]-[20]] because our results can applied to the case where $p_2(t) \neq 0$, $p(t) \neq 0$, $g(t,x(\tau_2(t))) \neq 0$ and $\alpha \neq \beta \neq \gamma$, specially the results obtained in [12] and [9] are considered as special cases of our results when taking $\alpha = \beta = \gamma$, $p(t) = p_2(t) = 0$ and $g(t,x(\tau_2(t))) = 0$ or either $g(t,x(\tau_2(t))) = 0$ or $f (t,x(\tau_1(t)))$ respectively. An example is given to illustrate our main result.