I have solved the following equation:
$2^x (2+\sqrt{3})^x - 2(1+\sqrt{3})^x = 2$
My approach:
since $2+\sqrt{3}=\frac{1}{2}(1+\sqrt{3})^2$ it follows that $(2+\sqrt{3})^x=\frac{1}{2^x}(1+\sqrt{3})^{2x}$ so our initial equation becomes $2^x \frac{1}{2^x}(1+\sqrt{3})^{2x} - 2(1+\sqrt{3})^x = 2 \Leftrightarrow $$ (1+\sqrt{3})^{2x} - 2(1+\sqrt{3})^x = 2$. By setting $t:= (1+\sqrt{3})^{x}$ we get the quadratic equation $t^2-2t-2=0\ (t>0) \Leftrightarrow t = 1+\sqrt{3} \Leftrightarrow $$(1+\sqrt{3})^x = 1+\sqrt{3} \Leftrightarrow x = 1$.
I would be grateful if someone could confirm this. Thanks.