Suppose the equation has rational roots:
4x^(2) + 5y^(2) = 2
then the question becomes does the equation:
where a,b,c are natural numbers has solution.
because 2c is even b must be even also, but then b=(4a-2c)/5 then 4a-2c is divisble by 5, i.e 4a-2c=0 or a and c are multiples of 5.
if b=0 then c=2a i.e (q2q1)^2=2(p1q2)^2 but this isn't possible cause sqrt(2) is irrational.
Now you need to check the option of multiples of 5 and get a contradiction, perhaps someone else with more intel in number theory will chip in with this question.