Question

for what values of k roots of
2x^2 + 4k x + 128 = 0 are imaginary

Answer 1

c'mon, man. I know that yo are aware of the discriminant for quadratics. Complex roots occur when it is negative. Thatis, when

(4k)^2 - 4*2*128 < 0
k^2 < 64
|k| < 8

confirm this by trying a couple of values.

Answer 2

by the way first divide the whole mess by 2 to simplify your life.