$$\frac{ 2+ \sqrt{ 3 } }{ 2- \sqrt{ 3 } } =A+B \sqrt{ 3 }$$
$B=-\frac{\sqrt{3}\left(A-4\sqrt{3}-7\right)}{3}$
$$\frac{\left(2+\sqrt{3}\right)\left(2+\sqrt{3}\right)}{\left(2-\sqrt{3}\right)\left(2+\sqrt{3}\right)}=A+B\sqrt{3}$$
$$\frac{\left(2+\sqrt{3}\right)\left(2+\sqrt{3}\right)}{2^{2}-\left(\sqrt{3}\right)^{2}}=A+B\sqrt{3}$$
$$\frac{\left(2+\sqrt{3}\right)\left(2+\sqrt{3}\right)}{4-3}=A+B\sqrt{3}$$
$$\frac{\left(2+\sqrt{3}\right)\left(2+\sqrt{3}\right)}{1}=A+B\sqrt{3}$$
$$\left(2+\sqrt{3}\right)\left(2+\sqrt{3}\right)=A+B\sqrt{3}$$
$$\left(2+\sqrt{3}\right)^{2}=A+B\sqrt{3}$$
$$4+4\sqrt{3}+\left(\sqrt{3}\right)^{2}=A+B\sqrt{3}$$
$$4+4\sqrt{3}+3=A+B\sqrt{3}$$
$$7+4\sqrt{3}=A+B\sqrt{3}$$
$$A+B\sqrt{3}=7+4\sqrt{3}$$
$$B\sqrt{3}=7+4\sqrt{3}-A$$
$$\sqrt{3}B=-A+4\sqrt{3}+7$$
$$\frac{\sqrt{3}B}{\sqrt{3}}=\frac{-A+4\sqrt{3}+7}{\sqrt{3}}$$
$$B=\frac{-A+4\sqrt{3}+7}{\sqrt{3}}$$
$$B=\frac{\sqrt{3}\left(-A+4\sqrt{3}+7\right)}{3}$$
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$A=-\sqrt{3}B+4\sqrt{3}+7$
