jmc

We see that $-2$ is not in the range of $f(x)=x^2+bx+2$ if and only if the equation $x^2+bx+2=-2$ has no real roots. We can re-write this equation as $x^2+bx+4=0$. The discriminant of this quadratic is $b^2-4\cdot 4=b^2-16$. The quadratic has no real roots if and only if the discriminant is negative, so $b^2-16<0$, or $b^2<16$. The set of values of $b$ that satisfy this inequality is $b\in \boxed{(-4,4)}$.