Consider the following quantity: $$ (1-p)^{n-2}-(1-q)^{n-2}+(n-2)q(1-q)^{n-3}-(n-2)p(1-p)^{n-3} $$ where $n\ge4$ is an integer, and $q-p\triangleq c$ is supposed to be fixed and in the interval $(0,1)$, i.e., the quantity can also be written as: $$ (1-p)^{n-2}-(1-p-c)^{n-2}+(n-2)(p+c)(1-p-c)^{n-3}-(n-2)p(1-p)^{n-3} $$
How can I show that this quantity is positive for $p\in (1/n-c/2,2/n-c/2)$?
Simulation shows that this quantity is decreasing in the given interval, so I tried taking derivative w.r.t. $p$, trying to show it's negative and solve the problem by plugging in $p=2/n-c/2$, but it gives a quantity similar to the quantity of interest, so either it doesn't really help, or I'm missing something here.
I have also tried using binomial formula to break up the parentheses, but couldn't get anything meaningful also.
Any suggestion and advice would be greatly appreciated!