Parametric curve question (determining unknown point)

cherry said:

Homework Statement: A curve given parametrically by (x, y, z) = (3 - t, -1 - 3t^2, 2t + 2t^3). There is a unique point P on the curve with the property that the tangent line at P passes through the point (2, 8, 12). What are the coordinates of point P?
Relevant Equations: (x, y, z) = (3 - t, -1 - 3t^2, 2t + 2t^3)

My work so far:
View attachment 338514

I am stuck because when I inputted the two possible values of t and k, neither solution worked. Where did I go wrong? Pointers would be appreciated! :)

cherry said:

I see where I went wrong and it turns out t = -1 and k = 2 is the correct solution.

##\qquad## !​

cherry said:

Where would I go from there to determine point P?