I know this is pretty basic, but I am unable to factor $2x^3+3x^2-8x+3$ without the use of trial/error. Even the rational roots theorem requires trial/error, as you need to implement the rational roots theorem to find at least one of the roots. The only other way (besides the rational root theorem) I found to factor this was to add two terms and subtract two terms. However, this is also pretty useless as it too requires number sense and trial/check. I know a solution to my problem would be graphing, but that would take too much time (even by finding the derivative and critical values) and I have no graphing calculator.
My question is, what method would I implement to factor without trial/error? Additionally, if this expression would be set equal to zero and by chance this expression has all complex/irrational roots ($x^3-2=0$), would this method work?