quick factoring question

1. I don't think you need to factor anything here.
2. Speed comes with practice
3. I fixed the typo in your first post (changed the + sign to -)

This is a tough equation, definitely designed to slow you down since even using the Quadratic formula it will take some time and few know that 729 is a square root of 27...

Your best bet is to think what 2 numbers are factors of 180 (but I would not want to do that on the real test)... I think the Quadratic formula is the best way to go and looking at 729 (the 9 is a giveaway that it is either 3 or 7 that will be involved).

bb is right. Use the quadratic formula if it takes a lot of time to proceed by hit and trial.

for hit and trial:

I haven't yet seen any GMAT quadratic question in the form of ax^2+bx+c=0 with fractions/irrational numbers as roots . Almost always the solution will be integers. I'll write my thought process; see if that helps you quicken your hit and trial: (obviously this will look way longer than what happens in my head :) )

r^2-3r-180=0

this is what I make out of 180 and 3 at the first instant:
looking for 2 numbers whose product is 180 and sum is -3.
sum is negative -> implies bigger of the absolute factors is negative and the smaller is positive.

the second step is:
I break 180 into the simplest factors I can see. 18 x 10 in this case.
if it doesn't happen for me (as in this case) i break each into 2 more (simple) factors. (6x3)x(5x2)

this does it for me. I know 12 and 15 are the numbers I'm looking for. The key is not to break 180 into its prime factors but two simpler composite factors.

I don't know whether this helps or not. I just meant to show that you can work on your hit and trial (it should not take a ton of time). Saying that, if you don't have enough time for your GMAT, it would be wise to use the quadratic formula.

Best