Developing
#(x+y√2)^2+(z+t√2)^2=x^2+z^2+2(y^2+t^2)+2(xy+zt) sqrt2 = 5+4√2#
so we have
#{(x^2+2y^2+z^2+2t^2=5),(xy+zt=2):}#
making now
#y = lambda x# and #t = mu z#
#{(x^2(1+2lambda^2)+z^2(1+2mu^2)=5),(x^2 lambda+z^2 mu = 2):}#
obtaining
#{(x^2 =((5 - 4 mu) mu-2)/((lambda - mu) ( 2 lambda mu-1)) ),(z^2=(2 + lambda (4 lambda-5))/((lambda - mu) (2 lambda mu-1))):}#
now
#(x/z)^2 = ((5 - 4 mu) mu-2)/(2 + lambda (4 lambda-5)) = -(4 mu^2-5 mu+2)/(4lambda^2-5lambda+2)#
and also
#4 mu^2-5 mu+2 > 0# and #4lambda^2-5lambda+2 > 0# for all #{mu, lambda} in QQ^2#
We conclude, as #(x/z)^2 < 0#, that the rational formulation is false.
There do not exist #{x,y,z,t} in QQ^4# such that
#(x+y√2)^2+(z+t√2)^2 = 5+4sqrt2#
