Pre cal f(x)=x^3+x^2+4x+4 solve with synthetic division and find all zeros of the function

The given polynomial is f(x) = x³ + x² + 4x + 4

Use the Remainder Theorem to identify the first zero.
From the Rational Zeros Theorem, possible zeros are +/- 1, +/- 2, +/- 4.
f(1) = 1 + 1 + 4 + 4 = 10 (not a zero).
f(-1) = -1 + 1 - 4 + 4 = 0  (This is a zero)

Therefore (x+1) is a factor of f(x).
Use Synthetic Division.

-1 | 1  1   4    4
        -1   0  -4
   
     1   0  4   0

There is no remainder, therefore
f(x) = (x+1)(x²+4)
Because x² = -4 yields x = 2i, -2i, there are a pair of conjugate complex zeros.

All the zeros are -1, 2i, -2i.

False because how does that work?