Example 2. Compare $\frac{23^{1088}+1}{23^{1880}+1}$ with $\frac{23^{1089}+1}{23^{1880}+1}$.
Solve by first comparing the quotient of $-\frac{x+1}{23}$ and $\frac{23 x+1}{23^{2} x+1}$.
$$
\begin{array}{l}
23 \frac{x+1}{x+1} \div \frac{23 x+1}{23^{2} x+1} \\
=\frac{(x+1)\left(23^{2} x+1\right)}{(23 x+1)^{2}} \\
=\frac{23^{2} x^{2}+\left(23^{2}+1\right) x+1}{23^{2} x^{2}+2 \times 23 x+1}>1 \text {. } \\
\end{array}
$$

When $x=23^{1988}$,
$$
\frac{23^{1988}+1}{23^{1989}+1}>\frac{23^{1989}+1}{23^{1980}+1}
$$