Final answer:
The only root of the equation x^3 + x^2 + 4x + 4 = 0 is x = -2.
Step-by-step explanation:
To find the roots of the equation x3 + x2 + 4x + 4 = 0, we can use the Rational Root Theorem to identify possible rational roots. The leading coefficient is 1, so the possible rational roots are factors of the constant term, which is 4. The factors of 4 are ±1, ±2, and ±4. By substituting these values into the equation, we find that the only rational root is x = -2.
Using synthetic division or long division, we can divide the given equation by (x + 2) to obtain the quadratic equation x2 - x + 2 = 0. We can then solve this quadratic equation using the quadratic formula or factoring.
The discriminant of the quadratic equation is negative, which means it has no real solutions. Therefore, the only root of the given cubic equation is x = -2.