$\displaystyle A=\left(\begin{array}{ccc}-3 & 1 & -1\\-7 & 5 & -1\\-6 & 6 & -2\end{array}\right)$find the definative polinomial?

if i go by definition

$\displaystyle |tI-A|=(t+3)[t^{2}-3t-4]+13t+20$

so i need to open the cols and find the roots of polinomial of power 3 which is alot of work

the answer in the book is $\displaystyle (t+2)^{2}(t-4)$

is there some other way to get this result

?