Explanation

(a)
$2\log_{e}x - \log_{e}\left\{ \left( 1 + \frac{1}{x} \right)x \right\} - \log_{e}\left\{ \left( 1 - \frac{1}{x} \right)x \right\}$
$= 2\log_{e}x - \left\{ \log_{e}\left( 1 + \frac{1}{x} \right) + \log_{e}x \right\} - \left\{ \log_{e}\left( 1 - \frac{1}{x} \right) + \log_{e}x \right\}$
$= - \left\{ \log_{e}\left( 1 + \frac{1}{x} \right) + \log_{e}\left( 1 - \frac{1}{x} \right) \right\} = 2\left\lbrack \frac{1}{2x^{2}} + \frac{1}{4x^{4}} + ... \right\}$
$\therefore$ The coefficients of $x^{- 4} = 2 \times \frac{1}{4} = \frac{1}{2}$