Question

Find the equation of the plane that passes through the point (-2,3,-1) and contains the line x(t)=-4+4t, y(t)=-4-3t, z(t)=-1-t

Answer 1

In your analysis, you considered that line joining any point (x, y, z) of the plane and the point
(6, 0, -2) is perpendicular to the given line which is not the case.

The simplest way will be to take any two points of the line and with (6, 0, -2) as the third point, find the equation of the plane.
t = 0 => (4, 3, 7) and
t = 1 => (2, 8, 11)
Find the equation of the plane containing points
(4, 3, 7), (2, 8, 11) and (6, 0, -2) using the formula in the form of a determinant.
Hope you can complete it from here.

Answer 2

I'm trying to find the equation of a place so wouldn't it be more useful to use points that are on the given line?
also, when plugging in 0 and 1 I did not get the same points as you... plugging in zero should result it (-4,-4,-1)