real roots

let f(x) = ax²+bx+c

a(70a - 12b + 2c)> 0

\Rightarrow a( f(-5)+f(-7) -4a)>0

\Rightarrow a >0 , and \;\;f(-5)+f(-7)>4a

or\;\; [ii] a <0 , and \;\;f(-5)+f(-7)<4a

now\; a >0 , and \;\;f(-5)+f(-7)>4a >0

it is an upward opening parabola , and here also there can be cases

i.e f(-5) and f(-7) both positive , or f(-5) > f(-7) , or f(-7) > f(-5)

for case [i i]

it wil be a downward opening parabola , and there will be cases

f(-5) and f(-7) both negative , or f(-5) < - f(-7)

now try to draw all the graphs possible from the above cases