a
(a) \({{5x + 6} \over {(2 + x)\,(1 - x)}} = {{{{ - 4} \over 3}} \over {2 + x}} + {{{{11} \over 3}} \over {1 - x}}\)

Rewriting the denominators for expressions, we get

= \({{{{ - 4} \over 3}} \over {2\left( {1 + {x \over 2}} \right)}} + {{{{11} \over 3}} \over {1 - x}} = {{ - 2} \over 3}{\left( {1 + {x \over 2}} \right)^{ - 1}} + {{11} \over 3}{(1 - x)^{ - 1}}\)

= \({{ - 2} \over 3}\left[ {1 - {x \over 2} + {{{x^2}} \over 4} - {{{x^3}} \over 8} + ...... + {{( - 1)}^n}{{{x^n}} \over {{2^n}}} + ......} \right]\,\)

\( + {{11} \over 3}[1 + x + {x^2} + ....... + {x^n} + .....]\)

The coefficient of \({x^n}\) in the given expression is

\({{ - 2} \over 3}{( - 1)^n}{1 \over {{2^n}}} + {{11} \over 3}\).