Explanation

(a)
Now, $\frac{1}{(1 + x)(1 + x^{2})} = \frac{A}{1 + x} + \frac{Bx + C}{1 + x^{2}}$
Where $Bx + C = f(x)$
$\Rightarrow 1 = A\left( 1 + x^{2} \right) + (Bx + C)(1 + x)$
On comparing the coefficient of $x^{2},\ x$ and constant terms, we get
$0 = A + B,\ \ 0 = B + C$ and $1 = A + C$
$\Rightarrow \ \ \ A = C = \frac{1}{2}$ and $B = - \frac{1}{2}$
$\therefore\ \ \ \frac{1}{(1 + x)(1 + x^{2})} = \frac{1}{2(1 + x)} + \frac{- \frac{x}{2} + \frac{1}{2}}{1 + x^{2}}$
$\therefore\ \ \ f(x) = - \frac{x}{2} + \frac{1}{2} = \frac{1 - x}{2}$