Let $\mathrm{\alpha },\mathrm{\beta },\mathrm{\gamma }$ be the roots of ${\mathrm{x}}^3+\mathrm{x}+10=0.$ Write ${\mathrm{\alpha }}_1=\frac{\mathrm{\alpha }+\mathrm{\beta }}{{\mathrm{\gamma }}^2},{\mathrm{\beta }}_1=\frac{\mathrm{\beta }+\mathrm{\gamma }}{{\mathrm{\alpha }}^2},{\mathrm{\gamma }}_1=\frac{\mathrm{\gamma }+\mathrm{\alpha }}{{\mathrm{\beta }}^2}.$ Then the value of $\left({\mathrm{\alpha }}_1^3+{\mathrm{\beta }}_1^3+{\mathrm{\gamma }}_1^3\right)-\frac{1}{10}\left({\mathrm{\alpha }}_1^2+{\mathrm{\beta }}_1^2+{\mathrm{\gamma }}_1^2\right)$ is

 $\mathrm{\alpha },\mathrm{\beta },\mathrm{\gamma }$ be the roots of ${\mathrm{x}}^3+\mathrm{x}+10=0.$ 
$\alpha +\beta +\gamma =0,$
 $$\begin{gathered} {\mathrm{\alpha }}_1=\frac{\mathrm{\alpha }+\mathrm{\beta }}{{\mathrm{\gamma }}^2}=\frac{-\gamma }{{\gamma }^2}=-\frac{1}{\gamma },{\mathrm{\beta }}_1=\frac{\mathrm{\beta }+\mathrm{\gamma }}{{\mathrm{\alpha }}^2}=\frac{-\alpha }{{\alpha }^2}=-\frac{1}{\alpha },{\mathrm{\gamma }}_1=\frac{\mathrm{\gamma }+\mathrm{\alpha }}{{\mathrm{\beta }}^2}=\frac{-\beta }{{\beta }^2}=-\frac{1}{\beta } \\ . \end{gathered}$$
 $$\begin{gathered} The equation whose roots are {\alpha }_1,{\beta }_1,{\gamma }_1 is \\ f\left(-\frac{1}{x}\right)=0\Rightarrow \frac{-1}{x^3}-\frac{1}{x}+10=0 \end{gathered}$$

$10x^3-x^2-1=0$

${\alpha }_1+{\beta }_1+{\gamma }_1=\frac{1}{10},{\alpha }_1{\beta }_1+{\beta }_1{\gamma }_1+{\gamma }_1{\alpha }_1=0,{\alpha }_1{\beta }_1{\gamma }_1=\frac{1}{10}$

$$\begin{gathered} {\mathrm{\alpha }}_1^2+{\mathrm{\beta }}_1^2+{\mathrm{\gamma }}_1^2={\left({\alpha }_1+{\beta }_1+{\gamma }_1\right)}^2-2\left({\alpha }_1{\beta }_1+{\beta }_1{\gamma }_1+{\gamma }_1{\alpha }_1\right)=\frac{1}{100} \\ \Rightarrow \boxed{{\mathrm{\alpha }}_1^2+{\mathrm{\beta }}_1^2+{\mathrm{\gamma }}_1^2=\frac{1}{100}} \end{gathered}$$

${\left({\mathrm{\alpha }}_1+{\mathrm{\beta }}_1+{\mathrm{\gamma }}_1\right)}^3={\mathrm{\alpha }}_1^3+{\mathrm{\beta }}_1^3+{\mathrm{\gamma }}_1^3+3\left(\left({\mathrm{\alpha }}_1+{\mathrm{\beta }}_1+{\mathrm{\gamma }}_1\right)\left({\mathrm{\alpha }}_1{\mathrm{\beta }}_1+{\mathrm{\beta }}_1{\mathrm{\gamma }}_1+{\mathrm{\gamma }}_1{\mathrm{\alpha }}_1\right)-,{\alpha }_1{\beta }_1{\gamma }_1\right)$

$\frac{1}{1000}={\mathrm{\alpha }}_1^3+{\mathrm{\beta }}_1^3+{\mathrm{\gamma }}_1^3+3\left(0-\frac{1}{10}\right)\Rightarrow \boxed{{\mathrm{\alpha }}_1^3+{\mathrm{\beta }}_1^3+{\mathrm{\gamma }}_1^3=\frac{1}{1000}+\frac{3}{10}}$

$\left({\mathrm{\alpha }}_1^3+{\mathrm{\beta }}_1^3+{\mathrm{\gamma }}_1^3\right)-\frac{1}{10}\left({\mathrm{\alpha }}_1^2+{\mathrm{\beta }}_1^2+{\mathrm{\gamma }}_1^2\right)=\frac{1}{1000}+\frac{3}{10}-\frac{1}{10}\left(\frac{1}{100}\right)=\frac{3}{10}$