Answer:
(i) No
(ii) The line and curve always intersect in two places
Explanation:
You want to know if there are values of k such that the curve y = (k+3)x²-3x lies completely above the line y = x+k. And you want to know the relationship between the line and the curve.
(i) Discriminant
The number of points of intersection between the line and the curve can be found by considering the discriminant of the quadratic equation for the difference of the y-values.
The line and curve will intersect when their y-values are equal:
(k +3)x² -3x = x +k
(k +3)x² -4x -k = 0 . . . . . compare to ax²+bx+c=0
The discriminant is the value of b²-4ac, so is ...
d = (-4)² -4(k+3)(-k) = 16 +4k(k -3)
Writing this in vertex form, we have ...
4(k² -3k) +16 = d
4(k² -3k +2.25) +16 -4(2.25) = d . . . . . "complete the square"
4(k -1.5)² +7 = d
The square is always positive, and it is added to a positive value (7), so the discriminant d is always positive. This means there are always two points of intersection between the curve and the line.
(ii) Relationship
As we saw above, the line and curve will always intersect at two points. This means there are always parts of the curve both above and below the line.
__
Additional comment
For the value k=-3, the "curve" becomes a line with slope -3 through the origin. It will intersect the line y=x-3 at one point. It seems reasonable to exclude that case from consideration.