a and b

[man111]

find [tex]a[/tex] and [tex]b[/tex] for which [tex]\displaystyle -5\leq \frac{x^2+ax+b}{x^2+2x+3}\leq 4[/tex] is true

[grav]

The denominator is always positive. So multiplying and rearranging we get two inequalities that must be true for all x. Thus you get two inequalities for the discriminants, which are quadratic in 'a' and linear in 'b'. Hence the solutions, of each inequality, consist of all points on one of the sides of each of those two parabolas. So you need to find the intersection of the corresponding regions, which is routine.