Example 10 If the equation
$$
\left(x^{2}-1\right)\left(x^{2}-4\right)=k
$$

has four non-zero real roots, and the four points corresponding to them on the number line are equally spaced, find the value of $k$.
Let $x^{2}=y$. Then the original equation becomes
$$
y^{2}-5 y+4-k=0 \text {. }
$$

By the given conditions, this equation has two distinct positive roots $\alpha, \beta$, without loss of generality, let $\alpha>\beta$.
Thus, the four real roots of the original equation are $\pm \sqrt{\alpha}, \pm \sqrt{\beta}$.
Since these four points corresponding to the roots are equally spaced on the number line, we have
$$
\sqrt{\alpha}-\sqrt{\beta}=\sqrt{\beta}-(-\sqrt{\beta}) .
$$

Therefore, $\alpha=9 \beta$.
Also, by Vieta's formulas, we know
$$
\alpha+\beta=5, \alpha \beta=4-k \text {. }
$$

Thus, $\alpha=\frac{9}{2}, \beta=\frac{1}{2}, k=\frac{7}{4}$.