Finding eigenvalues from characteristic polynomial

Question

I am finding it extremely hard to find the eigenvalues after finding the characteristic polynomial.

For example (instead of $\lambda$ I will use $x$) I have:

$-x^3+x^2+16x+20=0$, how do i find the eigenvalues? From the book, it says to use the factors of the constant, in this case the constant is $20$; and the one factors that does equals to $0$ are $5$ and $-2$. So how do I go about finding it?

I tried to take $x-5$ factor of $-x^3+x^2+16x+20$, but I cant seem to do it right.

Answer 1

Note that $x^3-x^2-16x-20=(x-5)(x^2+4x+4)=(x-5)(x+2)^2$.

Answer 2

Recall that if $f(a)=0$ then there is a $q(x)$ that $f(x)=(x-a)q(x)$. in your question $-5^3+5^2+16.5+20=0$.