I am worried that you have created q2=Root[ ,2] without a detailed understanding of the numbering of Root. I would guess that by your choosing to number your Root[ ,2] you may have forced Mathematica to assume things about your root that are incompatible with you then imposing additional constraints on b. If I were going to try this I would tell Mathematica what the constraints were, look at the results and try to decide which of those was the one I wanted.

If I ignore all that and just try

q2 = Root[(48 a^2 + 16 b) #^4 - (40 a^3 + 168 a b) #^3 + (-45 a^4 + 
        225 a^2 b + 72 b^2) #^2 + (27 a^3 b - 162 a b^2) # + 27 b^3 &, 2];
Simplify[Reduce[q2<c<0 && 0<b<a^2 && a<0, b, Reals], q2<c<0 && 0<b<a^2 && a<0]

then I get

3 a<4 c && b > Root[-45 a^4 c^2 - 40 a^3 c^3 + 48 a^2 c^4 + (27 a^3 c +
225 a^2 c^2 - 168 a c^3 + 16 c^4) #1 + (-162 a c + 72 c^2) #1^2 + 27 #1^3 &, 1]

so, because of the b > Root, I believe it knows that b is real and there is a constraint on the value of b.

I really doubt this resolves your uncertainties about the result.

For your second question, about whether r1<a^2, I would try to give Simplify that question and provide all the domain information you have that is applicable as the second argument to Simplify. Unfortunately I can't tell exactly what that would be from your question.

I understand you are annoyed with Root objects.

If you could paste neatly formatted code into your post instead of pictures then others would be able to paste that code back into Mathematica and see if they could help find a way to make any progress with your problem.

I suspect it may be very difficult to make progress when your constructed Root objects are built on multivariate polynomials of other variables. If someone gave you a quartic equation that contained not just x and exact coefficients, but other variables and demanded to know exactly what the roots were, not using Mathematica, I'm not sure that you could provide any answer to this unless the quartic happened to be a very special form. You might try such an exercise and see if you can make progress. Anything you learn from that exercise might be helpful.

Sometimes Simplify can help with the output from Reduce, particularly if you give your assumptions as the second argument to Simplify. Thus

Simplify[Reduce[stuff], 0<b<a^2]

or even

Simplify[Reduce[stuff], 0<b<a^2 && c<0 && a<4*c/3 && b<a^2]

or similar techniques might really help.

Use this cautiously.