I had this on a test and couldn't figure it out... It was written like this:
$12p^2-75$
The answer has to be $(p+\_ ) (p-\_)$
There must be a positive and negative number when combined (addition or subtraction) to equal $12$ but when multiplied to equal $-75$.
You've greatly garbled the routine for factoring a trinomial $ap^2+bp+c$. The answer does not "have to be" $(p+\dots)(p-\dots)$. In fact it won't be if $a\neq 1$.
If you're trying to factor $ap^2+bp+c$, then you want to find two numbers which when added give you $b$ and when multiplied give you $ac$, and if you are successful you can find a factorization of the trinomial.
In your case, we don't need that process anyway, because there is a simpler path. You should start by factoring 3 out of both (always, always try to find common factors first!):
$12p^2-75=3(4p^2-25)$
Now the idea is to notice that the quantity in parentheses is a difference of squares. The difference of squares always factors this way $X^2-Y^2=(X-Y)(X+Y)$ no matter what $X$ and $Y$ are.
Can you see what $X$ and $Y$ are for $(4p^2-25)$? After you know, you can write $3(X-Y)(X+Y)$ and you will be done.