$x^2-2x+4=0$

and $\alpha$ and $\beta$ are roots of this equation so

${\alpha }^2=2\alpha -4, {\beta }^2=2\beta -4$

Now ${\alpha }^6={\left(2\alpha -4\right)}^3=8{\alpha }^3-48{\alpha }^2+96\alpha -64$

$=8\alpha \left(2\alpha -4\right)-48\left(2\alpha -4\right)+96\alpha -64$

$=16\left(2\alpha -4\right)-32\alpha +128=64=2^6$

Similarly ${\beta }^6=2^6$

Hence ${\alpha }^{12}+{\beta }^{12}=2^{12}+2^{12}$$=2^{13}$