x^2 - 3x + 1=0 (divide x)
x - 3 + (1/x) =0
x+ (1/x) = 3
x^2 + (1/x^2 ) = (x + (1/x) )^2 -2 =3^2 -2 =7
therefore,
x^2 + (1/x^2 ) + 3 = 7 + 3 =10