Hello!!! Could anyone help me to solve this exercise?

Using Stirling's formula show that( see the first attached file stirling1!!!), where S(x)=-xlnx-(1-x)ln(1-x), 0<=x<=1.

I used n!=e^(-n)n^(n+1/2)(2π)^(1/2)*(1+O(1/n)) and my result is (see the second attached file stirling2).

Is this equal to the result i have to show??(Worried)

Thank you!!!!