-x^2 + 6x + 20 = b + x^2 - 4x + 8 rearrange as
2x^2 - 10x + (b - 12 ) = 0
It will have no real roots when the discriminant is < 0
So
10^2 - 4(2) (b - 12) < 0
100 - 8b + 96 < 0
196 < 8b
8b > 196
b > 196/ 8
b > 49 / 2
It will have no real roots when b > 49 / 2
