Your part b should have the inequality sign when you get to the quadratic expression with k.

i.e (k - 6)(k + 2) < 0 and not

(k - 6)(k + 2) = 0

The discriminant < 0 as there are no real roots for 2x² + (k + 2)x + 2 = 0 , since it is > 0 from the previous part.