That's fine as far is goes, but of course is not a complete proof. I assume you have worked out a proof of the statement that the difference of two squared odd numbers must be divisible by 8, right?
I like it is a very interesting discussion. I think we can approach the question as under:
let a = 2m+1, b = 2n +1 and c= 2p +1
now discriminant of bx^2 + ax + c = 0 is D = a^2 - 4 bc = ( 2m+1)^2 - 4 ( 2n +1) ( 2p+1) = 4m^2 + 4m + 1- 4 [ 4 np + 2 (n+p) +1 ]
= 4m^2 + 4m - 4 [ 4 np + 2 (n+p) ] - 3 = 4 [ m^2 + m - 4 np - 2 (n+p) ] - 3
Now we observe that it is not a perfect square. Thus the roots will be irrational.