Please help me with this. I'm good with quadratic inequalities that I can factor, but I've never been able to wrap my head around ones that aren't factorable.
lim 0.1x^2+2.157x+20 = 203
x->33.33
If that limit is correct then the polynomial you will be working with is factorable!
If the limit as x goes to 33.33 of 0.1x^2+ 2.157x+ 20 is equal to 203, then 0.1x^2+ 2.157x+ 20- 203= 0.1x^2+ 2.157x- 183 would have to go to 0 and so must be factorable.
Using either "long division" or "synthetic division", divide 0.1x^2+ 2.157x- 183 by x- 33.33 to find the other factor.
(One difficulty is that the limit, as x goes to 33.33, is NOT exactly 203. I suspect you mean the limit as x goes to 100/3.)