Let $\mathrm{\alpha },\mathrm{\beta },\mathrm{\gamma }$ be the three roots of the equation ${\mathrm{x}}^3+\mathrm{bx}+\mathrm{c}=0.\mathrm{If}\mathrm{\beta \gamma }=1=-\mathrm{\alpha },$then ${\mathrm{b}}^3+2{\mathrm{c}}^3-3{\mathrm{\alpha }}^3-6{\mathrm{\beta }}^3-8{\mathrm{\gamma }}^3$ is equal to 

$\alpha , \beta , \gamma \text{ are Roots of }{x}^3+bx+c=0$
$\begin{array}{l}\beta \gamma =1=-\alpha \\ \Rightarrow \alpha =-1\\ \text{ and }\alpha +\beta +\gamma =0\\ \alpha \beta \gamma =-c\\ c={\alpha }^2=1\\ \beta +\gamma =1\\ \begin{array}{l}\alpha \beta +\beta \gamma +\gamma \alpha =b\\ \Rightarrow \alpha (\beta +\gamma )+\beta \gamma =b\\ \Rightarrow b=0\\ \Rightarrow \beta =-\omega ,\gamma =-{\omega }^2\\ \Rightarrow b^3+2c^3-3{\alpha }^3-6{\beta }^3-8{\gamma }^3\\ =0+2+3+6+8\\ =19\end{array}\end{array}$