y = (k+3)x² - 3x . . . . . . . . . [1]
y = x + k . . . . . . . . . [2]

x + k = (k+3)x² - 3x
(k+3)x² - 4x - k = 0 . . . . . . . . . [3]

for curve [1] to be completely above line [2] ,
the discriminant of [3] is less than zero (they dun touch each other) ;

i.e.
16 + 4k(k+3) < 0
4 + k² + 3k < 0
k² + 3k + 4 < 0

now observe:
k² + 3k + 4
= (k + 3/2)² + 4 - 9/4
= (k + 1.5)² + 7/4

since
(k + 1.5)² is always +ve for any real values of k ; &
(k + 1.5)² + 7/4 is always +ve for any real values of k

so (k + 1.5)² + 7/4 cannot be less than zero for any real values of k

k² + 3k + 4 < 0 is invalid ;

hence,
there cannot exist any real values of k , for curve [1] to lie completely above line [2] .

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