Continued fractions with rational functions Express the following rational function in continued-fraction form:
$${4x^2+3x-7\over 2x^3+x^2-x+5}$$
The answer is :
$${4 \over 2x- \frac{1}{2}} + { \frac{23}{8} \over x-\frac{63}{92}}-{\frac{406}{529} \over x+\frac{33}{23}}\tag{inline continued fraction}$$
which means
$$
\cfrac{4}{2x- \frac{1}{2}+\cfrac{\frac{23}{8}}{x-\frac{63}{92}-\cfrac{\frac{406}{529}}{x+\frac{33}{23}}}}
$$

Here is the solution using integer coefficients
$$
\begin{align}
\cfrac{4x^2+3x-7}{2x^3+x^2-x+5}
&=\cfrac1{\cfrac{2x^3+x^2-x+5}{4x^2+3x-7}}\\
&=\cfrac{8}{4x-1+\cfrac{23x+33}{4x^2+3x-7}}\\
&=\cfrac{8}{4x-1+\cfrac{529}{92x-63-\cfrac{1624}{23x+33}}}\\
\end{align}
$$
The book's answer can be shown to be equal by cancelling fractions.
