Consider the multiplication operator $L_A:\mathbb{R}^4\to\mathbb{R}^4$ where $$A=\left[\begin{array}{cccc} -3 &4 &-4 &4\cr 10 &12 &-2 &25\cr 14 &9 &2 &23\cr -2 &-7 &3 &-12 \end{array}\right].$$ Let $W$ be the invariant subspace generated by $L_A$ and the vector $\left(1,1,-1,-1\right)$.
a. Find the characteristic polynomial $g(t)$ of the restriction $L_A | W$.
$g(t) =$
b. We know that $g(t)$ divides the characteristic polynomial $f(t)$ of $L_A$. Find the quotient polynomial $q(t):=f(t)/g(t)$.
$q(t) =$