#2x^2-8x+5# is of the form #ax^2+bx+c# with #a=2#, #b=-8# and #c=5#.
This has discriminant given by the formula:
#Delta = b^2-4ac = (-8)^2 - (4xx2xx5) = 64 - 40 = 24#
#Delta > 0# implies that #2x^2-8x+5 = 0# has two distinct real roots and therefore two linear factors with real coefficients.
Unfortunately, #Delta# is not a perfect square, so those coefficients are not rational.
The roots of #2x^2-8x+5 = 0# are given by the formula:
#x = (-b+-sqrt(Delta))/(2a) = (8 +- sqrt(24))/4#
#= 2+- sqrt(2^2*6)/4 = 2+-(2sqrt(6))/4 = (4+-sqrt(6))/2#
For symmetry I would like to multiply both sides of this by #sqrt(2)# to get:
#sqrt(2)x = (4sqrt(2)+-sqrt(2)sqrt(6))/2#
#= (4sqrt(2)+-sqrt(2)sqrt(2)sqrt(3))/2#
#= (4sqrt(2)+-2sqrt(3))/2 = 2sqrt(2)+-sqrt(3)#
and we find:
#2x^2-8x+5 = (sqrt(2)x-2sqrt(2)+sqrt(3))(sqrt(2)x-2sqrt(2)-sqrt(3))#
