Example 12 Express $\frac{2 x^{2}+2 x+13}{(x-2)\left(x^{2}+1\right)^{2}}$ as partial fractions.
Solution: Let $N=\frac{2 x^{2}+2 x+13}{(x-2)\left(x^{2}+1\right)^{2}}$
$$
\equiv \frac{A}{x-2}+\frac{B x+C}{x^{2}+1}+\frac{D x+E}{\left(x^{2}+1\right)^{2}} \text {. }
$$

By comparing coefficients, we get $A=1, B=-1$,
$$
C=-2, D=-3, E=-4 \text {. }
$$

Therefore, $N=\frac{1}{x-2}+\frac{-x-2}{x^{2}+1}+\frac{-3 x-4}{\left(x^{2}+1\right)^{2}}$.