In this paper, we study the oscillation problem of the following higher-order neutral differential equation with positive and negative coefficients and mixed arguments $$z^{(n)}(t)+q_1(t)|x(t-{sigma}_1)|^{alpha-1}x(t-{sigma}_1)+q_2(t)|x(t-{sigma}_2)|^{eta-1}x(t-{sigma}_2)=e(t)$$, where $t{geq}t_0$, $z(t)=x(t)-p(t)x(t-{ au})$ with $p(t)$ > 0, ${eta}>1>{alpha}>0$, ${ au}$, ${sigma}_1$ and ${sigma}_2$ are real numbers. Without imposing any restriction on ${ au}$, we establish several oscillation criteria for the above equation in two cases: (i) $q_1(t){leq}0$, $q_2(t)>0$, ${sigma}_1{geq}0$ and ${sigma}_2{leq}{ au}$; (ii) $q_1(t){geq}0$, $q_2(t)<0$, ${sigma}_1{geq}{ au}$ and ${sigma}_2{leq}0$. As an interesting application, our results can also be applied to the following higher-order differential equation with positive and negative coefficients and mixed arguments $$x^{(n)}(t)+q_1(t)|x(t-{sigma}_1)|^{alpha-1}x(t-{sigma}_1)+q_2(t)|x(t-{sigma}_2)|^{eta-1}x(t-{sigma}_2)=e(t)$$. Two numerical examples are also given to illustrate the main results.