2. Given $a=\sqrt{3}-1$. Then $a^{2012}+2 a^{2011}-2 a^{2010}=$
2. 0 .

Notice that $a^{2}=(\sqrt{3}-1)^{2}=4-2 \sqrt{3}$. Then $a^{2}+2 a-2=0$.
Therefore, $a^{2012}+2 a^{2011}-2 a^{2010}$ $=a^{2010}\left(a^{2}+2 a-2\right)=0$.