1. The problem asks for the values of $k$ such that the quadratic equation $$x^2 + kx + k^2 = 0$$ has complex (non-real) roots. 2. Recall that a quadratic equation $ax^2 + bx + c = 0$ has complex roots if its discriminant is less than zero: $$\Delta = b^2 - 4ac < 0.$$ 3. In our equation, $a=1$, $b=k$, and $c=k^2$, so the discriminant is: $$\Delta = k^2 - 4(1)(k^2) = k^2 - 4k^2 = -3k^2.$$ 4. For the roots to be complex (not real), we require: $$-3k^2 < 0.$$ 5. Since $k^2 \geq 0$ for all real $k$, $-3k^2 < 0$ if and only if $k^2 > 0$, meaning: $$k \neq 0.$$ 6. Therefore, the roots are complex if $k$ is any real number except zero. 7. Checking the options: - (a) $\mathbb{R} - \{0\}$ matches this condition. - (b), (c), and (d) exclude additional values or restrict $k$ to positive or negative intervals. **Final answer:** $k \in \mathbb{R} - \{0\}$.