The equation of the lines has to be determined that have a slope 2 and intersect y = x(x - 6) once, twice and never.
The general equation of a line with slope 2 is y = 2x + k.
If the line intersects the given curve:
2x + k = x(x - 6)
=> x^2 - 6x - 2x - k = 0
=> x^2 - 8x - k = 0
This has one solution when (-8)^2 = 4*(-k)*1
=> 64 = -4k
=> k = -16
It has no real solutions when 64 < -4k
=> 64 > 4k
=> -16 > k
It has two solutions for all values of k > -16
The lines y = 2x + k intersect y = x(x - 6) once if k = -16, twice if k > -16. There is no intersection if k < -16