5. Let $k$ be a real number, and $\alpha, \beta$ be the roots of the equation $x^{2} + k x - 1 = 0$. If $(|\alpha| - \beta)(|\beta| - \alpha) \geqslant 1$, then the range of $k$ is ( ).
(A) $k \geqslant \sqrt{\sqrt{5}-2}$
(B) $k \leqslant \sqrt{\sqrt{5}-2}$
(C) $k \leqslant -\sqrt{\sqrt{5}-2}$ or $k \geqslant \sqrt{\sqrt{5}-2}$
(D) $k \leqslant -\sqrt{\sqrt{5}-2}$
5.A.

Since $\Delta=k^{2}+4>0$, therefore, $\alpha, \beta$ are two distinct real roots. By Vieta's formulas, we have
$$
\alpha+\beta=-k, \alpha \beta=-1 .
$$

Since the left side of the inequality in the problem is symmetric with respect to $\alpha, \beta$, we can assume without loss of generality that $\alpha < \beta$. Therefore,
$$
\begin{array}{l}
\left\{\begin{array}{l}
k>0, \\
k^{2}(k^{2}+4) \geqslant 1
\end{array} \Rightarrow \left\{\begin{array}{l}
k>0, \\
k^{2} \geqslant \sqrt{5}-2
\end{array}\right.\right. \\
\Rightarrow k \geqslant \sqrt{\sqrt{5}-2} \text {. }
\end{array}
$$