Let tangent at  $({\alpha }_1,{\beta }_1)$  on the curve  $y=x^4-2x^2-x$ touch the curve again at  $({\alpha }_2,{\beta }_2)$, then the value of  $\left|{\alpha }_1\right|+\left|{\alpha }_2\right|+\left|{\beta }_1\right|+\left|{\beta }_2\right|$  is equal to

Let  $y=mx+c$ be the tangent  $\Rightarrow x^4-2x^2-x-(mx+c)={(x^2+ax+b)}^2$
 $\Rightarrow a=0,b=-1,m=-1,c=-1$
$\Rightarrow x+y+1=0$  is tangent to the curve at  $(-1,0)$  and  $(1,-2)$
$\Rightarrow \left|{\alpha }_1\right|+\left|{\alpha }_2\right|+\left|{\beta }_1\right|+\left|{\beta }_2\right|=4$