We have \begin{align*} &\left|\sqrt{x^2+2x+5}-\sqrt{x^2-4x+8}\right|<3 \\ \iff &2x^2-2x+13-2\sqrt{(x^2+2x+5)(x^2-4x+8)}< 9 \\ \iff &(x^2-x+2)^2 < (x^2+2x+5)(x^2-4x+8) \\ \iff &4< 40. \quad \blacksquare \end{align*}