The sum of the co-efficient of all even degree terms in x in the...

${\left(x + \sqrt{{x}^{3} - 1}\right)}^{6} + {\left(x - \sqrt{{x}^{3} - 1}\right)}^{6}$

$= 2\left({}^{6}{C}_{0}{x}^{6}{+}^{6}{C}_{2}{x}^{4}\left({x}^{3} - 1\right){+}^{6}{C}_{4}{x}^{2}{\left({x}^{3} - 1\right)}^{2}{+}^{6}{C}_{6}{\left({x}^{3} - 1\right)}^{3}\right)$

$= 2\left({}^{6}{C}_{0}{x}^{6}{+}^{6}{C}_{2}\left({x}^{7} - {x}^{4}\right){+}^{6}{C}_{4}{x}^{2}\left({x}^{6} - 2{x}^{3} + 1\right){+}^{6}{C}_{6}\left({x}^{9} - 3{x}^{6} + 3{x}^{3} - 1\right)\right)$

$∴$ Sum of the coefficient of even power of $x$

$= 2\left(6{C}_{0}{-}^{6}{C}_{2}{+}^{6}{C}_{4}{+}^{6}{C}_{4} - 3{-}^{6}{C}_{6}\right)$

$= 2\left(1 - 15 + 15 + 15 - 3 - 1\right) = 24$