### Instruction: Find all roots of the polynomial $x^3+x^2-4x-4$. Enter your answer as a list of numbers separated by commas.### Response: To find the roots of a cubic polynomial, I can try to factor it into a product of a linear term and a quadratic term.
To do this, I can use the rational root theorem, which says that any rational root of the polynomial must be a factor of the constant term divided by a factor of the leading coefficient.
In this case, the constant term is -4 and the leading coefficient is 1, so the possible rational roots are $\pm 1, \pm 2, \pm 4$.
I can test each of these candidates by plugging them into the polynomial and checking if the result is zero.
If I find one that works, I can use synthetic division or long division to find the corresponding quadratic factor.
Let me start with 1.
If I plug in 1, I get $1^3+1^2-4(1)-4=1+1-4-4=-6$, which is not zero.
So 1 is not a root.
Next, let me try -1.
If I plug in -1, I get $(-1)^3+(-1)^2-4(-1)-4=-1+1+4-4=0$, which is zero.
So -1 is a root.
That means I can factor out $(x+1)$ from the polynomial.
I can use synthetic division to do this.
I write -1 on the left and the coefficients of the polynomial on the right:

-1 | 1  1  -4  -4
   |    -1   0   4
   |________________
     1  0  -4   0
The bottom row gives me the coefficients of the quadratic factor.
So I have $x^3+x^2-4x-4=(x+1)(x^2-4)$.
Now I can use the difference of squares formula to factor the quadratic term further.
I get $x^3+x^2-4x-4=(x+1)(x-2)(x+2)$.
This means that the other roots are 2 and -2, since they make the linear factors zero.
Therefore, the list of all roots is -2, -1, 2.