Answer
a) $x^{2}+\frac{A}{x-1}+\frac{B}{(x-1)^2}$
b) $\frac{A}{x}+\frac{Bx+C}{x^{2}+x+1}$
Work Step by Step
a) $\frac{x^4-2x^{3}+x^{2}+2x-1}{x^{2}-2x+1}$ = $x^{2}+\frac{2x-1}{x^{2}-2x+1}$ = $x^{2}+\frac{2x-1}{(x-1)^2}$ = $x^{2}+\frac{A}{x-1}+\frac{B}{(x-1)^2}$
b) $\frac{x^2-1}{x^{3}+x^{2}+x}$ = $\frac{x^2-1}{x(x^{2}+x+1)}$ = $\frac{A}{x}+\frac{Bx+C}{x^{2}+x+1}$
