$$\frac{2+\sqrt{7}}{2-\sqrt{7}}=a+b\sqrt{7}$$
$b=-\frac{\sqrt{7}a}{7}-\frac{11\sqrt{7}}{21}-\frac{4}{3}$
$$\frac{\left(2+\sqrt{7}\right)\left(2+\sqrt{7}\right)}{\left(2-\sqrt{7}\right)\left(2+\sqrt{7}\right)}=a+b\sqrt{7}$$
$$\frac{\left(2+\sqrt{7}\right)\left(2+\sqrt{7}\right)}{2^{2}-\left(\sqrt{7}\right)^{2}}=a+b\sqrt{7}$$
$$\frac{\left(2+\sqrt{7}\right)\left(2+\sqrt{7}\right)}{4-7}=a+b\sqrt{7}$$
$$\frac{\left(2+\sqrt{7}\right)\left(2+\sqrt{7}\right)}{-3}=a+b\sqrt{7}$$
$$\frac{\left(2+\sqrt{7}\right)^{2}}{-3}=a+b\sqrt{7}$$
$$\frac{4+4\sqrt{7}+\left(\sqrt{7}\right)^{2}}{-3}=a+b\sqrt{7}$$
$$\frac{4+4\sqrt{7}+7}{-3}=a+b\sqrt{7}$$
$$\frac{11+4\sqrt{7}}{-3}=a+b\sqrt{7}$$
$$\frac{-11-4\sqrt{7}}{3}=a+b\sqrt{7}$$
$$-\frac{11}{3}-\frac{4}{3}\sqrt{7}=a+b\sqrt{7}$$
$$a+b\sqrt{7}=-\frac{11}{3}-\frac{4}{3}\sqrt{7}$$
$$b\sqrt{7}=-\frac{11}{3}-\frac{4}{3}\sqrt{7}-a$$
$$\sqrt{7}b=-a-\frac{4\sqrt{7}}{3}-\frac{11}{3}$$
$$\frac{\sqrt{7}b}{\sqrt{7}}=\frac{-a-\frac{4\sqrt{7}}{3}-\frac{11}{3}}{\sqrt{7}}$$
$$b=\frac{-a-\frac{4\sqrt{7}}{3}-\frac{11}{3}}{\sqrt{7}}$$
$$b=-\frac{\sqrt{7}a}{7}-\frac{11\sqrt{7}}{21}-\frac{4}{3}$$
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$a=-\sqrt{7}b-\frac{4\sqrt{7}}{3}-\frac{11}{3}$
