Solution:
We observe that
$$\left|\sqrt{x^{2}+4x+8}-\sqrt{x^{2}+8x+17}\right|=\left|\sqrt{(x-(-2))^{2}+(0-2)^{2}}-\sqrt{(x-(-4))^{2}+(0-1)^{2}}\right|$$
is the absolute difference of the distances from the point $P(x, 0)$ in the $xy$-plane to the points $A(-2,2)$ and $B(-4,1)$.
By the Triangle Inequality, $|PA-PB| \leq |AB|$ and the equality occurs exactly when $P$ lies on the line passing through $A$ and $B$, but not between them.
If $P, A, B$ are collinear, then $(x-(-2))/(0-2)=((-4)-(-2))/(1-2)$. This gives $x=-6$, and as $-6<-4<-2$,
$$\left|\sqrt{(-6)^{2}+4(-6)+8}-\sqrt{(-6)^{2}+8(-6)+17}\right|=|\sqrt{20}-\sqrt{5}|=\sqrt{5}$$
is the largest possible value of the expression.