4. 41 Solve the equation $5 x^{2}+x-x \cdot \sqrt{5 x^{2}-1}-2=0$.
［Solution］ Transform the original equation into
$$\left(5 x^{2}-1\right)-x \cdot \sqrt{5 x^{2}-1}+(x-1)=0,$$

Factorizing gives
$$\left(\sqrt{5 x^{2}-1}-x+1\right)\left(\sqrt{5 x^{2}-1}-1\right)=0$$

From $\sqrt{5 x^{2}-1}-x+1=0$, we get $x_{1}=\frac{1}{2}, x_{2}=-1$.
From $\sqrt{5 x^{2}-1}-1=0$, we get $x_{3,4}= \pm \frac{\sqrt{10}}{5}$.
Upon verification, $x_{1}=\frac{1}{2}, x_{2}=-1$ are extraneous roots. Therefore, the roots of the original equation are $\pm \frac{\sqrt{10}}{5}$.