If x=ω-ω^ 2 -2, then the value of x^ 4 +3 x^ 3 +2 x^ 2 -11 x-6 is

We have, $x=\omega-\omega^{2}-2$ or $x+2=\omega-\omega^{2}$ Squaring, $x^{2}+4 x+4=\omega^{2}+\omega^{4}-2 \omega^{3}=\omega^{2}+\omega^{3} \omega \cdot-2 \omega^{3}=\omega^{2}+\omega-2 \quad\left[\omega^{3}=1\right]=-1-2=-3 \Rightarrowx^{2}+4 x+7=0$ Dividing $x^{4}+3 x^{3}+2 x^{2}-11 x-6$ by $x^{2}+4 x+7$ we get x^{4}+3 x^{3}+2 x^{2}-11 x-6=\left(x^{2}+4 x+7\right)\left(x^{2}-x-1\right)+1=(0)\left(\mathrmx^{2}-\mathrm{x}-1\right)+1=0+1=1