Is there 2 ways to write a parabolic equation, without facing your teacher's fury?

Question

This is me being extremely stupid, but a parabola with a vertex $(2,6)$ and $x$-intercepts of $(-4,0),(8,0)$ can be described with the equation $y=\frac{-(x+4)(x-8)}{6}$(my answer) which would also be the same as $\frac{-(x-2)^2}{6}+6$(the goddamn textbook's answer), yes? So if I were to write that on a test, let's say, would I be wrong since the textbook answers are the exact opposite of mine or are my answer(s) acceptable. I Am Confusion, even though this is extremely basic, any help welcome.

Answer 1

Both ways you wrote are correct, and both are ways to present a parabola. Will your teacher mark off points? I don't know.

Mathematically, they are both correct. Personally, I would write it as

$$ y = -\frac{1}{6}x^2 + \frac{2}{3}x + \frac{16}{3} $$

but again, these are all correct!