- -3<x<-2: (x+2)(x+3)<0; x^2+5x+6<0. Put x=-2.5, then we have (-0.5)(0.5)=-0.25 which is less than 0 and satisfies this inequality. Put x=-3: the polynomial is zero and the inequality fails as expected; put x=-2 and the inequality fails as expected, but between the limits the inequality holds.
- 1<x<3: (x-1)(x-3)<0; x^2-4x+3<0. When x=2 we have (1)(-1)=-1<0 is true. So the inequality holds between the limits. Outside the limits the inequality fails as expected.
The two inequalities are x^2+5x+6 and x^2-4x+3. These quadratics can be combined by multiplication:
x^4-4x^3+ 3x^2
+5x^3-20x^2+15x
+ 6x^2 -24x+18
x^4 +x^3-11x^2 - 9x+18 is a quartic that fulfils both inequalities.