$$\frac{\sqrt{2}+\sqrt{3}}{3\sqrt{2}-2\sqrt{3}}=a+b\sqrt{6}$$
$b=-\frac{\sqrt{6}a}{6}+\frac{\sqrt{6}}{3}+\frac{5}{6}$
$$\frac{\left(\sqrt{2}+\sqrt{3}\right)\left(3\sqrt{2}+2\sqrt{3}\right)}{\left(3\sqrt{2}-2\sqrt{3}\right)\left(3\sqrt{2}+2\sqrt{3}\right)}=a+b\sqrt{6}$$
$$\frac{\left(\sqrt{2}+\sqrt{3}\right)\left(3\sqrt{2}+2\sqrt{3}\right)}{\left(3\sqrt{2}\right)^{2}-\left(-2\sqrt{3}\right)^{2}}=a+b\sqrt{6}$$
$$\frac{\left(\sqrt{2}+\sqrt{3}\right)\left(3\sqrt{2}+2\sqrt{3}\right)}{3^{2}\left(\sqrt{2}\right)^{2}-\left(-2\sqrt{3}\right)^{2}}=a+b\sqrt{6}$$
$$\frac{\left(\sqrt{2}+\sqrt{3}\right)\left(3\sqrt{2}+2\sqrt{3}\right)}{9\left(\sqrt{2}\right)^{2}-\left(-2\sqrt{3}\right)^{2}}=a+b\sqrt{6}$$
$$\frac{\left(\sqrt{2}+\sqrt{3}\right)\left(3\sqrt{2}+2\sqrt{3}\right)}{9\times 2-\left(-2\sqrt{3}\right)^{2}}=a+b\sqrt{6}$$
$$\frac{\left(\sqrt{2}+\sqrt{3}\right)\left(3\sqrt{2}+2\sqrt{3}\right)}{18-\left(-2\sqrt{3}\right)^{2}}=a+b\sqrt{6}$$
$$\frac{\left(\sqrt{2}+\sqrt{3}\right)\left(3\sqrt{2}+2\sqrt{3}\right)}{18-\left(-2\right)^{2}\left(\sqrt{3}\right)^{2}}=a+b\sqrt{6}$$
$$\frac{\left(\sqrt{2}+\sqrt{3}\right)\left(3\sqrt{2}+2\sqrt{3}\right)}{18-4\left(\sqrt{3}\right)^{2}}=a+b\sqrt{6}$$
$$\frac{\left(\sqrt{2}+\sqrt{3}\right)\left(3\sqrt{2}+2\sqrt{3}\right)}{18-4\times 3}=a+b\sqrt{6}$$
$$\frac{\left(\sqrt{2}+\sqrt{3}\right)\left(3\sqrt{2}+2\sqrt{3}\right)}{18-12}=a+b\sqrt{6}$$
$$\frac{\left(\sqrt{2}+\sqrt{3}\right)\left(3\sqrt{2}+2\sqrt{3}\right)}{6}=a+b\sqrt{6}$$
$$\frac{3\left(\sqrt{2}\right)^{2}+5\sqrt{2}\sqrt{3}+2\left(\sqrt{3}\right)^{2}}{6}=a+b\sqrt{6}$$
$$\frac{3\times 2+5\sqrt{2}\sqrt{3}+2\left(\sqrt{3}\right)^{2}}{6}=a+b\sqrt{6}$$
$$\frac{6+5\sqrt{2}\sqrt{3}+2\left(\sqrt{3}\right)^{2}}{6}=a+b\sqrt{6}$$
$$\frac{6+5\sqrt{6}+2\left(\sqrt{3}\right)^{2}}{6}=a+b\sqrt{6}$$
$$\frac{6+5\sqrt{6}+2\times 3}{6}=a+b\sqrt{6}$$
$$\frac{6+5\sqrt{6}+6}{6}=a+b\sqrt{6}$$
$$\frac{12+5\sqrt{6}}{6}=a+b\sqrt{6}$$
$$2+\frac{5}{6}\sqrt{6}=a+b\sqrt{6}$$
$$a+b\sqrt{6}=2+\frac{5}{6}\sqrt{6}$$
$$b\sqrt{6}=2+\frac{5}{6}\sqrt{6}-a$$
$$\sqrt{6}b=-a+\frac{5\sqrt{6}}{6}+2$$
$$\frac{\sqrt{6}b}{\sqrt{6}}=\frac{-a+\frac{5\sqrt{6}}{6}+2}{\sqrt{6}}$$
$$b=\frac{-a+\frac{5\sqrt{6}}{6}+2}{\sqrt{6}}$$
$$b=-\frac{\sqrt{6}a}{6}+\frac{\sqrt{6}}{3}+\frac{5}{6}$$
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$a=-\sqrt{6}b+\frac{5\sqrt{6}}{6}+2$
