The partial fraction decomposition of $\frac{10}{(x-1)(x+1)}$ can be written in the form of $\frac{f(x)}{x-1}+\frac{g(x)}{x+1}$, where
\[
\begin{array}{l}
f(x)= \\
g(x)=
\end{array}
\]
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Therefore, $f(x)=5$ and $g(x)=-5$.

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Step 1 ：Multiplying both sides by $(x-1)(x+1)$ gives us $$10 = f(x)(x+1)+g(x)(x-1).$$

Step 2 ：Setting $x=1$ gives $10=2f(1)$, and so $f(1)=5$.

Step 3 ：Setting $x=-1$ gives $10=-2g(-1)$, and so $g(-1)=-5$.

Step 4 ：Therefore, $f(x)=5$ and $g(x)=-5$.