Solution:
We have the following algebra:
$$\begin{aligned} \frac{x^{4}-x^{2}}{x^{6}+2 x^{3}-1} & = \frac{x-\frac{1}{x}}{x^{3}+2-\frac{1}{x^{3}}} \\ & = \frac{x-\frac{1}{x}}{\left(x-\frac{1}{x}\right)^{3}+2+3\left(x-\frac{1}{x}\right)} \\ & \leq \frac{x-\frac{1}{x}}{3\left(x-\frac{1}{x}\right)+3\left(x-\frac{1}{x}\right)} = \frac{1}{6} \end{aligned}$$
where $\left(x-\frac{1}{x}\right)^{3}+1+1 \geq 3\left(x-\frac{1}{x}\right)$ in the denominator was deduced by the AM-GM inequality. As a quick check, equality holds where $x-\frac{1}{x}=1$ or when $x=\frac{1+\sqrt{5}}{2}$.