Find the largest possible value of the expression $\left|\sqrt{x^{2}+4 x+8}-\sqrt{x^{2}+8 x+17}\right|$ where $x$ is a real number.
We observe that

$$
\left|\sqrt{x^{2}+4 x+8}-\sqrt{x^{2}+8 x+17}\right|=\left|\sqrt{(x-(-2))^{2}+(0-2)^{2}}-\sqrt{\left.(x-(-4))^{2}+(0-1)^{2}\right)}\right|
$$

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 \(\frac{x-(-2)}{0-2} = \frac{(-4)-(-2)}{1-2}\). 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.