$x^3-6x+5=0$  

$x^3\underset{-6x}{\underbrace{-x-5x}}+5=0$   

$x\left(x^2-1\right)-5\left(x-1\right)=0$  

$x\left(x+1\right)\left(\mathbf{x}-1\right)-5\left(\mathbf{x}-1\right)=0$  

$\left(\mathbf{x}-1\right)\left(x\left(x+1\right)-5\right)=0$  

$\left(x-1\right)\underset{\left(\star \right)}{\underbrace{\left(x^2+x-5\right)}}=0$  

Wyznaczamy pierwiastki równania  $\left(\star \right)$ . 

$x^2+x-5=0$   

$\Delta =1^2-4\cdot 1\cdot \left(-5\right)=1+20=21$  

$\sqrt{\Delta }=\sqrt{21}$  

$x_1=\frac{-1-\sqrt{21}}{2\cdot 1}=\frac{-1-\sqrt{27}}{2}=-\frac{1+\sqrt{21}}{2}$  

$x_2=\frac{-1+\sqrt{21}}{2\cdot 1}=\frac{-1+\sqrt{21}}{2}=-\frac{1-\sqrt{21}}{2}$  

Zatem: 

$\left(x+\frac{1+\sqrt{21}}{2}\right)\left(x+\frac{1-\sqrt{21}}{2}\right)=0$  

Wracając do wyjściowego równania mamy: 

$\left(x-1\right)\left(x+\frac{1+\sqrt{21}}{2}\right)\left(x+\frac{1-\sqrt{21}}{2}\right)=0$  

Zatem: 

$x_1=-\frac{1+\sqrt{21}}{2},x_2=1,x_3=-\frac{1-\sqrt{21}}{2}$