Help please!

The only root of x^3-x+2 is   -1.5214  (by graphing)

-1.5214^6 - ( -1.5214)^2 +4(-1.5214) = 4.00

Let t be a root of

$f(x) = x^3 - x + 2$.

Evaluate

$t^6 - t^2 + 4t$.

1.

$\begin{array}{|rcll|} \hline f(t) = t^3 - t + 2 &=& 0 \\ t^3 - t + 2 &=& 0 \\ \mathbf{t^3} &\mathbf{=}&\mathbf{t-2} \\ \hline \end{array}$

2.

$\begin{array}{|rcll|} \hline t^6 - t^2 + 4t &=& (t^3)^2 - t^2 + 4t \quad | \quad \mathbf{t^3=t-2} \\ &=& (t-2)^2 - t^2 + 4t \\ &=& t^2-4t + 4 - t^2 + 4t \\ \mathbf{t^6 - t^2 + 4t} &\mathbf{=}&\mathbf{4} \\ \hline \end{array}$