For how many value(s) of b does the equation $x^3+b^2x^2+2x+3=0$ have integral solutions?
This equation can either have one or three real (not necessarily integral) solutions. Using Descarte's rule of signs I found that it has either one or three negative roots. I also know this functions is increasing so I am assuming it only has one negative real root.
However, how do I relate this to b?
I assume $b$ is supposed to be an integer?
The easiest approach to this problem is probably to use the usual method for finding the rational roots to an integer polynomial -- the rational root theorem. All of the rational roots are of the form $p/q$ where $p$ divides the constant term and $q$ divides the leading coefficient. (of course, you only need the integer roots... but all of the candidate roots the rational root theorem suggests are already integers)