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The partial fraction decomposition of $\cfrac{x^2}{(x-1)(x^2+2x+1)}$ is given by:

$\cfrac{1/4}{x-1}+\cfrac{3x+1}{4(x^2+2x+1)}$

$\cfrac{1/4}{x-1}+\cfrac{-1/2}{(x+1)^2}$

$\cfrac{1/4}{x-1}+\cfrac{-1/2}{(x+1)^2}+\cfrac{3/4}{x+1}$

$\cfrac{x+1}{2(x^2-1)}+\cfrac{1}{2(x+1)}$