Given equation,

$${x^2} + (2i - 1) = 0$$

$$ \Rightarrow {x^2} = 1 - 2i$$

Let $$\alpha$$ and $$\beta$$ are the two roots of the equation.

As, we know roots of a equation satisfy the equation so

$${\alpha ^2} = 1 - 2i$$

and $${\beta ^2} = 1 - 2i$$

$$\therefore$$ $${\alpha ^2} = {\beta ^2} = 1 - 2i$$

$$\therefore$$ $$|{\alpha ^2}| = \sqrt {{1^2} + {{( - 2)}^2}} = \sqrt {15} $$

Now, $$|{\alpha ^8} + {\beta ^8}|$$

$$|{\alpha ^8} + {\alpha ^8}|$$

$$ = 2|{\alpha ^8}|$$

$$ = 2|{\alpha ^2}{|^4}$$

$$ = 2{\left( {\sqrt 5 } \right)^4}$$

$$ = 2 \times 25$$

$$ = 50$$