The following is a guessed solution because the equation of the curve has been assumed. However, the method is valid and can be applied even if the assumption is incorrect.
Parabola y=2x²-4x+8, dy/dx=4x-4, the slope of the tangent line, and the line must have the same slope to be a tangent, so, y=(4a-4)x if a is the point of contact.
(4a-4)a=2a²-4a+8 is where the line meets the curve.
So 4a²-4a=2a²-4a+8, 2a²=8, a²=4, a=±2. There are two tangent points.
The y coordinates are: 8-8+8=8 and 8+8+8=24, making the points of contact (2,8) and (-2,24) and the tangent lines y=4x and y=-12x.
So if y=kx represents the line, the two values for k are 4 and -12.