### 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: I notice that this is a cubic polynomial, so it might have up to three roots.
I also notice that the constant term is -4, so I wonder if there are any integer roots that are factors of -4, such as -1, -2, -4, 1, 2, or 4.
I can try plugging in each of these values and see if they make the polynomial zero.
If I plug in x = -1, I get (-1)^3 + (-1)^2 - 4(-1) - 4, which is -1 + 1 + 4 - 4, which is zero.
So x = -1 is one root of the polynomial.
I can use synthetic division to divide the polynomial by x + 1 and get a quadratic quotient.
I write the coefficients of the polynomial in a row: 1, 1, -4, -4.
I write -1 below the first coefficient and bring down the 1.
I multiply -1 by 1 and write the result below the second coefficient: -1.
I add 1 and -1 and write the result below the line: 0.
I multiply -1 by 0 and write the result below the third coefficient: 0.
I add -4 and 0 and write the result below the line: -4.
I multiply -1 by -4 and write the result below the fourth coefficient: 4.
I add -4 and 4 and write the result below the line: 0.
I see that the quotient is 1, 0, -4, with a remainder of 0.
This means that the polynomial is equal to (x + 1)(x^2 - 4).
I can factor the quadratic term further as (x + 1)(x + 2)(x - 2).
This means that the other two roots of the polynomial are x = -2 and x = 2.
So the complete list of roots is -1, -2, 2.