Bài giải:

P = $\left( \frac{1}{x-1}+\frac{x}{{{x}^{3}}-1}.\frac{{{x}^{2}}+x+1}{x+1} \right):\frac{2x+1}{{{x}^{2}}+2x+1}$

P = $\left( \frac{1}{x-1}+\frac{x}{(x-1)(x+1)} \right):\frac{2x+1}{{{(x+1)}^{2}}}$

P = $\frac{x+1+x}{(x-1)(x+1)}.\frac{{{(x+1)}^{2}}}{2x+1}$

P = $\frac{2x+1}{(x-1)(x+1)}.\frac{{{(x+1)}^{2}}}{2xx+1}$

P = $\frac{x+1}{x-1}$

Thay x = $\frac{2017}{2015}$ vào biểu thức P ta có:

P = $\frac{\frac{2017}{2015}+1}{\frac{2017}{2015}-1}$ = $\frac{4032}{2015}.\frac{2015}{2}$ = 2016

Vậy đáp án đúng là: B.