check your arithmetic.
for det(tI - A), i get (t+3)(t^2 - 3t - 4) + t - 4 = t^3 - 12t - 16.
to factor t^3 - 12t - 16, you might try guessing rational (and thus integer) roots first, that is: t = -1,1,-2,2,-4,4,-8,8,-16,16.
on the 3rd try, you would find that (-2)^3 - 12(-2) - 16 = -8 + 24 -16 = 0, so you would know that t+2 is a factor.
division by t+2 will give you a quadratic, to which you could apply the quadratic formula (unless you could factor it by inspection).