Is there a way to show $\frac{1646-736\sqrt{5}}{2641-1181\sqrt{5}}=\frac{17+15\sqrt{5}}{7+15\sqrt{5}}$ without multiplying large numbers?

Note that \begin{eqnarray*} \frac{1646-736\sqrt{5}} {2641-1181\sqrt{5}} =1 -5\frac{199-89\sqrt{5}}{2641-1181\sqrt{5}} \end{eqnarray*} and \begin{eqnarray*} \frac{89\sqrt{5}-199}{2641-1181\sqrt{5}}=\frac{2}{7+15 \sqrt{5}}. \end{eqnarray*}