A theory of equation question from my exam paper

Question

Consider The equation $x^3+3x^2+3x+3=0$ Then the sum of it's non-real roots is A) is equal to $0$ B)lies in $0$ and $1$ C)lies in $-1$ and $0$ D)Greter that $1$

Which one is correct , plz explain briefly;

Answer 1

HINT:

$$x^3+3x^2+3x+3=0\implies (x+1)^3=-2$$

Do you about the roots of $y^3=a$ where $a$ is a real number?

If no, please have a look into this problem

Answer 2

As the eqn shows the imaginary roots are $x=-s\omega -1,-s\omega^2 -1$ where $s$ is the real cube root of $2$.Now summing them we get the sum is $s-2$.Now it remains to show that $-1<s-2<0\implies 1<s<2\implies 1<s^3<8\implies 1<2<8$ which is true so the option C) is correct.