Note that $$\sqrt{x^2 - 4x + 5} + \sqrt{x^2 +4x +8} = \sqrt{(x-1)^2 + (0-1)^2} + \sqrt{(x+2)^2 + (0+2)^2}$$ This equals the distance between points $(x, 0)$ and $(2, 1)$ plus the distance between $(x, 0)$ and $(-2, -2)$. Clearly, it will reach minimal when these three points are collinear in which case the sum is just the distance between $(2, 1)$ and $(-2, -2)$, i.e. $\boxed{5}$.