24. Given that
$$
S=\left|\sqrt{x^{2}+4 x+5}-\sqrt{x^{2}+2 x+5}\right|,
$$
for real values of $x$, find the maximum value of $S^{4}$.
24. Answer: 4
$$
S=\left|\sqrt{(x+2)^{2}+(0-1)^{2}}-\sqrt{(x+1)^{2}+(0-2)^{2}}\right| .
$$

Let $P=(x, 0), A=(-2,1)$ and $B=(-1,2)$, then $S$ represents the difference between the lengths $P A$ and $P B . S$ is maximum when the points $P, A$ and $B$ are collinear and that occurs when $P=(-3,0)$. So
$$
S=\left|\sqrt{(-1)^{2}+(1)^{2}}-\sqrt{(-2)^{2}+(0-2)^{2}}\right|=|\sqrt{2}-2 \sqrt{2}| .
$$

Thus the maximum value of $S^{4}=4$.