Answer
$a.\displaystyle \qquad \frac{A}{x}+\frac{B}{x^{2}}+\frac{Cx+D}{1+x^{2}}$
$b.\displaystyle \qquad 1+\frac{A}{x}+\frac{B}{x-1}+\frac{C}{x-2}$
Work Step by Step
$a.$
Factor the denominator:
$x^{2}+x^{4}=x^{2}(1+x^{2})$
thre are repeated factors and we also have a quadratic factor.
$\displaystyle \frac{1}{x^{2}+x^{4}}=\frac{A}{x}+\frac{B}{x^{2}}+\frac{Cx+D}{1+x^{2}}$
$b.$
The denominator and numerator have same degrees. Either use polynomial division or:
$\displaystyle \frac{(x^{3}-3x^{2}+2x)+3x^{2}-2x+1}{x^{3}-3x^{2}+2x}=1+\frac{3x^{2}-2x+1}{x^{3}-3x^{2}+2x}$
Factor the denominator.
$x^{3}-3x^{2}+2x=x(x^{2}-3x+2)=x(x-1)(x-2)$
$\displaystyle \frac{3x^{2}-2x+1}{x^{3}-3x^{2}+2x}=\frac{A}{x}+\frac{B}{x-1}+\frac{C}{x-2}$
$\displaystyle \frac{x^{3}+1}{x^{3}-3x^{2}+2x}=\frac{A}{x}+\frac{B}{x-1}+\frac{C}{x-2}$
