(a) Use the binomial expansion for \(\left( 1 - \frac{1}{2x} \right)^2\):

\(= 1 - 2 \times \frac{1}{2x} + \left( \frac{1}{2x} \right)^2\)

\(= 1 - \frac{1}{x} + \frac{1}{4x^2}\).

(b) Use the binomial expansion for \((1 + 2x)^6\):

First four terms are:

\(1 + \binom{6}{1} (2x) + \binom{6}{2} (2x)^2 + \binom{6}{3} (2x)^3\)

\(= 1 + 12x + 60x^2 + 160x^3\).

(c) Multiply the expansions from (a) and (b) and find the coefficient of \(x\):

\((1 - \frac{1}{x} + \frac{1}{4x^2})(1 + 12x + 60x^2 + 160x^3)\)

Coefficient of \(x\) is:

\(1 \times 12 + (-1) \times 60 + \frac{1}{4} \times 160\)

\(= 12 - 60 + 40 = -8\).