Example 5 Convert the mixed periodic continued fraction

into a quadratic irrational.

Translate the text above into English, please keep the original text's line breaks and format, and output the translation result directly.
Explanation: Let $y=[4,5, \overline{2,3}]$, with the repeating part being $x=[\overline{2,3}]$. Then,
$$
y=4+\frac{1}{5}+\frac{1}{x}=[4,5, x] \text {. }
$$

Thus, we have Table 5.
Table 5
\begin{tabular}{|c|c|c|c|c|c|}
\hline$a_{n}$ & & & 4 & 5 & $x$ \\
\hline$p_{n}$ & 0 & 1 & 4 & 21 & $21 x+4$ \\
\hline$q_{n}$ & 1 & 0 & 1 & 5 & $5 x+1$ \\
\hline
\end{tabular}

From Example 4, we know that $x=\frac{3+\sqrt{15}}{3}$. Therefore,
$$
y=\frac{21 x+4}{5 x+1}=\frac{75+21 \sqrt{15}}{18+5 \sqrt{15}}=\frac{75-\sqrt{15}}{17} .
$$