Which form of this quadratic do i use to solve intercept and range? So my equation is:
$-2x^2 + 4x + 30 = 0$

If I use this form to look at my y intercept, it will be 30.
However, once I simplify it to: $x^2 - 2x - 15$, then my y intercept will be $-15$. Which one do I use? Inclusively, I was taught that if the quadratic is positive, it is in a U shape, and if it is negative, it is an upside-down U shape.

How do I know which form to use to solve these questions?
Your equation is not $-2x^2 + 4x + 30 = 0$, but $f(x)=-2x^2 + 4x + 30 = -2(x^2-2x-15)$. You must find the $y$ intercept of a function, rather than an equation. To find the $y$ intercept, you need to let $x=0.$

$$f(0)=-2(0^2-2(0)-15)=-2(0-0-15)=-2(-15)=30$$

Thus, the $y$ intercept still occurs at $(0,30)$

When $-2x^2 + 4x + 30 = 0$, you are saying that you are trying to find where $f(x)=0$, and not a general function, $f(x)$. Note that $f(x)=-2x^2 + 4x + 30 \neq x^2-2x-15 $ 

Please comment if what I have said does not make sense.
It does not simplify to $x^{2} - 2x - 15$, you have to continue to account for the $-2$ you factored out:

$$-2x^{2} + 4x + 30 = -2(x^{2} - 2x - 15) = 0 $$

Evaluating at $x = 0$, the intercept remains $(0,30)$.