a
(a) \({{{x^4} + 24{x^2} + 28} \over {{{({x^2} + 1)}^3}}} = {{{A_1}x + {B_1}} \over {{x^2} + 1}} + {{{A_2}x + {B_2}} \over {{{({x^2} + 1)}^2}}} + {{{A_3}x + {B_3}} \over {{{({x^2} + 1)}^3}}}\)

==>\({x^4} + 24{x^2} + 28 = ({A_1}x + {B_1})\,{({x^2} + 1)^2}\)

\( + ({A_2}x + {B_2})\,({x^2} + 1) + ({A_3}x + {B_3})\)

Putting \(x = i,\,5 = {A_3}i + {B_3}\)\( \Rightarrow \) \({A_3} = 0,\,{B_3} = 5\)

Equating different powers of \(x\),

\(0 = {A_1},\,{B_1} = 1,\,2{A_1} + {A_2} = 0 \Rightarrow {A_2} = 0\)

\(2{B_1} + {B_2} = 24 \Rightarrow {B_2} = 22\).

\(\therefore\) Partial fraction = \({1 \over {{x^2} + 1}} + {{22} \over {{{({x^2} + 1)}^2}}} + {5 \over {{{({x^2} + 1)}^3}}}\).