If $x = \frac{\sqrt{111}-1}{2}$, calculate $(2x^{5} + 2x^{4} - 53x^{3} - 57x + 54)^{2004}$.

Need long division

Divide $$2x^5+2x^4-53x^3-57x+54$$ by $2x^2+2x-55$ to express

$$2x^5+2x^4-53x^3-57x+54=q(x)\cdot(2x^2+2x-5)+r(x)$$ where $q(x)$ is the quotient and $r(x)$ is the remainder.

$\implies2x^5+2x^4-53x^3-57x+54=r(x)$ as $2x^2+2x-5=0$

Here $r(x)=-1$