Would this be sufficient as a proof?

[sooyong94]

Homework Statement

2. Homework Equations
Vieta's formula, quadratic discriminant

3. The Attempt at a Solution

 

[Ssnow]

suggestion for the first part: since $\alpha,\beta$ are solutions your equation as also a second form $(x-\alpha)(x-\beta)=0$ so you can expand this and proceed comparing with $ax^{2}+2bx+c=0$ ...
for the second part you must to look to the $\Delta$ of the equation $ax^{2}+2mbx+nc=0$ and to prove that is $\geq 0$ (observe that $b^{2}-ac\geq 0$ by assumption) ...

Your attempt at a solution is very long, I hope with my suggestions you will able to simplify ...

 

[haruspex]

I was a bit confused because the order of the solution attachments does not match the order of the questions. The last attachment appears to address the first question. As Ssnow writes, it is simpler to start with factorisation.

The first attached solution seems unrelated to the posted questions.

The second attached solution does seem to be related to the second posted question, but exactly how is unclear.

The third attached solution does address the second question, but the logic is backwards. You are asked to show that the equation has real roots, so you should not start with "if it has real roots". You could start with "it will only have real roots if..."