Question

$x^{2} + x + 1 = 0$

$\Rightarrow x = \omega$ or $\omega^{2}$

$${\therefore\alpha = \omega\&\beta = \omega^{2} }{= \left( \omega + \omega^{2} \right)^{4} + \left( \omega^{2} + \omega^{4} \right)^{4} + \left( \omega^{3} + \omega^{6} \right)^{4} + \ldots + \left( \omega^{25} + \omega^{50} \right)^{4} }{= \left\lbrack \left( \omega + \omega^{2} \right)^{4} + \left( \omega^{2} + \omega^{4} \right)^{4} + \left( \omega^{4} + \omega^{8} \right)^{4} + \ldots. + \right.\ \left. \ \left( \omega^{25} + \omega^{50} \right)^{4} \right\rbrack + \left\lbrack \left( \omega^{3} + \omega^{6} \right)^{4} + \left( \omega^{6} + \omega^{12} \right)^{4} + \right.\ \left. \ \left( \omega^{9} + \omega^{18} \right)^{4} + \ldots + \left( \omega^{24} + \omega^{48} \right)^{4} \right\rbrack }{= \underset{17\text{~times~}}{\overset{\lbrack 1 + 1 + 1\ldots\ldots + 1\rbrack}{︸}} + \underset{8\text{~times~}}{\overset{\left\lbrack (1 + 1)^{4} + (1 + 1)^{4} + \ldots..(1 + 1)^{4} \right\rbrack}{︸}} }{= 17 + 128 }{= 145}$$