$$\frac{5+2\sqrt{3}}{7+4\sqrt{3}}=a+b\sqrt{3}$$
$b=-\frac{\sqrt{3}\left(a+6\sqrt{3}-11\right)}{3}$
$$\frac{\left(5+2\sqrt{3}\right)\left(7-4\sqrt{3}\right)}{\left(7+4\sqrt{3}\right)\left(7-4\sqrt{3}\right)}=a+b\sqrt{3}$$
$$\frac{\left(5+2\sqrt{3}\right)\left(7-4\sqrt{3}\right)}{7^{2}-\left(4\sqrt{3}\right)^{2}}=a+b\sqrt{3}$$
$$\frac{\left(5+2\sqrt{3}\right)\left(7-4\sqrt{3}\right)}{49-\left(4\sqrt{3}\right)^{2}}=a+b\sqrt{3}$$
$$\frac{\left(5+2\sqrt{3}\right)\left(7-4\sqrt{3}\right)}{49-4^{2}\left(\sqrt{3}\right)^{2}}=a+b\sqrt{3}$$
$$\frac{\left(5+2\sqrt{3}\right)\left(7-4\sqrt{3}\right)}{49-16\left(\sqrt{3}\right)^{2}}=a+b\sqrt{3}$$
$$\frac{\left(5+2\sqrt{3}\right)\left(7-4\sqrt{3}\right)}{49-16\times 3}=a+b\sqrt{3}$$
$$\frac{\left(5+2\sqrt{3}\right)\left(7-4\sqrt{3}\right)}{49-48}=a+b\sqrt{3}$$
$$\frac{\left(5+2\sqrt{3}\right)\left(7-4\sqrt{3}\right)}{1}=a+b\sqrt{3}$$
$$\left(5+2\sqrt{3}\right)\left(7-4\sqrt{3}\right)=a+b\sqrt{3}$$
$$35-6\sqrt{3}-8\left(\sqrt{3}\right)^{2}=a+b\sqrt{3}$$
$$35-6\sqrt{3}-8\times 3=a+b\sqrt{3}$$
$$35-6\sqrt{3}-24=a+b\sqrt{3}$$
$$11-6\sqrt{3}=a+b\sqrt{3}$$
$$a+b\sqrt{3}=11-6\sqrt{3}$$
$$b\sqrt{3}=11-6\sqrt{3}-a$$
$$\sqrt{3}b=-a+11-6\sqrt{3}$$
$$\frac{\sqrt{3}b}{\sqrt{3}}=\frac{-a+11-6\sqrt{3}}{\sqrt{3}}$$
$$b=\frac{-a+11-6\sqrt{3}}{\sqrt{3}}$$
$$b=\frac{\sqrt{3}\left(-a+11-6\sqrt{3}\right)}{3}$$
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$a=-\sqrt{3}b+11-6\sqrt{3}$
