$$\frac{3-2\sqrt{5}}{6-\sqrt{5}}=a+b\sqrt{5}$$
$b=-\frac{\sqrt{5}a}{5}+\frac{8\sqrt{5}}{155}-\frac{9}{31}$
$$\frac{\left(3-2\sqrt{5}\right)\left(6+\sqrt{5}\right)}{\left(6-\sqrt{5}\right)\left(6+\sqrt{5}\right)}=a+b\sqrt{5}$$
$$\frac{\left(3-2\sqrt{5}\right)\left(6+\sqrt{5}\right)}{6^{2}-\left(\sqrt{5}\right)^{2}}=a+b\sqrt{5}$$
$$\frac{\left(3-2\sqrt{5}\right)\left(6+\sqrt{5}\right)}{36-5}=a+b\sqrt{5}$$
$$\frac{\left(3-2\sqrt{5}\right)\left(6+\sqrt{5}\right)}{31}=a+b\sqrt{5}$$
$$\frac{18-9\sqrt{5}-2\left(\sqrt{5}\right)^{2}}{31}=a+b\sqrt{5}$$
$$\frac{18-9\sqrt{5}-2\times 5}{31}=a+b\sqrt{5}$$
$$\frac{18-9\sqrt{5}-10}{31}=a+b\sqrt{5}$$
$$\frac{8-9\sqrt{5}}{31}=a+b\sqrt{5}$$
$$\frac{8}{31}-\frac{9}{31}\sqrt{5}=a+b\sqrt{5}$$
$$a+b\sqrt{5}=\frac{8}{31}-\frac{9}{31}\sqrt{5}$$
$$b\sqrt{5}=\frac{8}{31}-\frac{9}{31}\sqrt{5}-a$$
$$\sqrt{5}b=-a-\frac{9\sqrt{5}}{31}+\frac{8}{31}$$
$$\frac{\sqrt{5}b}{\sqrt{5}}=\frac{-a-\frac{9\sqrt{5}}{31}+\frac{8}{31}}{\sqrt{5}}$$
$$b=\frac{-a-\frac{9\sqrt{5}}{31}+\frac{8}{31}}{\sqrt{5}}$$
$$b=-\frac{\sqrt{5}a}{5}+\frac{8\sqrt{5}}{155}-\frac{9}{31}$$
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$a=-\sqrt{5}b-\frac{9\sqrt{5}}{31}+\frac{8}{31}$
