$a)$

$\left(1\right)a^2-1\ne 0\Rightarrow a\in \mathbb{R}\backslash \left\{-1,1\right\}\Rightarrow st\left(w\right)=3$

$\left(2\right)a=1\Rightarrow a^2-1=0,-\left(a+1\right)=-2\ne 0\Rightarrow st\left(w\right)=1$

$\left(3\right)a=-1\Rightarrow a^2-1=0,-\left(a+1\right)=0\Rightarrow st\left(w\right)=0$