fac = (x^4) - 8  (x^2) + 4 // Factor

fac[[1]] /. {x -> (Sqrt[4 + 2 (Sqrt[ 3])])}

fac[[1]] /. {x -> (Sqrt[4 + 2 (Sqrt[ 3])])} // RootReduce

(Sqrt[4 + 2 (Sqrt[ 3])]) // RootReduce