The denominator in the given expression $\frac{x^2}{(x-1)^2(x+1)}$ has a repeated linear factor $(x-1)^2$ and a distinct linear factor $(x+1)$. Therefore, the partial fraction decomposition will be of the form:
\[
\frac{x^2}{(x-1)^2(x+1)} = \frac{A}{x-1} +
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