The value of the expression  $x^4-8x^3+18x^2-8x+2whenx=2+\sqrt{3}$

Given  $x=2+\sqrt{3},x-2=\sqrt{3}squaringonbothsides$

$$\begin{gathered} {(x-2)}^2=3\Rightarrow x^2-4x+4=3 \\ {x}^2-4x+1=0\mathrm{........}\left(1\right) \\ \begin{array}{l}\Rightarrow {(x-2)}^2=3\\ squaringonbothsides\begin{array}{l}\Rightarrow {(x-2)}^2=3\end{array}=9\end{array} \\ \begin{array}{l}{x}^4-4x^3\left(2\right)\text{I}+6x^2{\left(2\right)}^2-4x{\left(2\right)}^2+{\left(2\right)}^4=9\\ \Rightarrow x^4-8x^3+24x^2-32x+16=9\\ \Rightarrow x^4-8x^3+18x^2-8x+2+6x^2-24x+6-1=0\\ \Rightarrow (x^4-8x^3+18x^2-8x+2)+6(x^2-4x+1)=1\\ from\left(1\right)x^4-8x^3+18x^2-8x+2+6\left(0\right)=1\\ x^4-8x^3+18x^2-8x+2=1\end{array} \end{gathered}$$