8. The product of the maximum and minimum values of the function $f(x)=\frac{x-x^{3}}{1+2 x^{2}+x^{4}}$ is . $\qquad$
8. $-\frac{1}{16}$.

Let $x=\tan \theta\left(\theta \in\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)\right)$. Then
$$
\begin{array}{c}
f(x)=\frac{\tan \theta\left(1-\tan ^{2} \theta\right)}{\left(1+\tan ^{2} \theta\right)^{2}} \\
=\sin \theta \cdot \cos \theta \cdot \cos 2 \theta \\
=\frac{1}{4} \sin 4 \theta \in\left[-\frac{1}{4}, \frac{1}{4}\right] .
\end{array}
$$