$$\begin{gathered} \mathrm{If} \mathrm{\alpha }, \mathrm{\beta } \mathrm{are} \mathrm{the} \mathrm{roots} \mathrm{of} \mathrm{the} \mathrm{equation} \\ 1+\mathrm{x}+{\mathrm{x}}^2=0, \mathrm{then} \left(2-\mathrm{\alpha }\right)\left(2-\mathrm{\beta }\right)\left(2-{\mathrm{\alpha }}^{10}\right)\left(2-{\mathrm{\alpha }}^{20}\right)= \end{gathered}$$

$\mathrm{\alpha }, \mathrm{\beta }$ are roots of $1+\mathrm{x}+{\mathrm{x}}^2=0$

$\begin{array}{rcl} & \Rightarrow & \mathrm{\alpha }=\mathrm{w}, \mathrm{\beta }={\mathrm{w}}^2\end{array}$

$\begin{array}{rcl} & & (2-\mathrm{\alpha }) (2-\mathrm{\beta }) (2-{\mathrm{\alpha }}^{10}) (2-{\mathrm{\alpha }}^{20})\end{array}$

$\begin{array}{rcl} & & (2-\mathrm{w}) (2-{\mathrm{w}}^2) (2-{\mathrm{w}}^{10}) (2-{\mathrm{w}}^{20})\end{array}$

$\begin{array}{rcl} & & (4-2{\mathrm{w}}^2-2\mathrm{w}+{\mathrm{w}}^3) (4-2{\mathrm{w}}^{20}-2{\mathrm{w}}^{10}+{\mathrm{w}}^{30})\end{array}$

$\begin{array}{rcl} & & (4-2({\mathrm{w}}^2+\mathrm{w})+1) (4-2({\mathrm{w}}^2+\mathrm{w})+{\left({\mathrm{w}}^3\right)}^{10})\end{array}$

$\begin{array}{rcl} & & (4-2(-1)+1) (4-2(-1)+1)\end{array}$

$\begin{array}{rcl} & & \\ \left(7\right) \left(7\right)& =& 49\end{array}$