Math Help - equation of tangent plane to a surface

1. equation of tangent plane to a surface

I managed to do the first part okay ---- said some stuff about 3x^2 and y^2 term, its not linear etc..

but Im stuck in the part in red. Is it supposed to be something about a normal vector? How do i know what is wrong? and what mistake was made?

I calculated the equation of the tangent plane and it is: z=12(x−2)−6(y−3)-1

and the tangent plane has normal vector (12,-6,-1) at (2,3) ----- could it be something to do with the part circled in red?

2. Yes, "Answer 1" is clearly wrong because it is not linear and so not the equation of any plan.

Now, "Answer 2" is a plane- it is linear parametric equations with two parameters.
The simplest answer is that when x= 2 and y= 3, z= 8- 9= -1. But there are no values of $\lambda$ and $\mu$ that make $(\lambda, \mu, 12\lambda- 6\mu)= (2, 3, -1)$. It should be easy to see that and that means that the given plane does NOT include that point.

3. Thanks, and for part 2 --- im meant to say what mistake the student made (b) and how to fix it (c). How would i find the "precise" mistake the student made, and how can i correct "answer 2"?

4. for the 2nd part, i got (lamba, mu, 12lamba-6mu-7)

So can i say the student forgot to put/calculate in the -7 in the parametric form? At a glance, how would i know the student got it wrong? Because he didnt put in the -7?

5. Yes, that's good. As I said before, the simplest way to see that the second answer was wrong was to note that it did not include the point (2, 3, -1). As for "what mistake the student made", there could have been many! But the most likely was just that he forgot the "-7".