1. We are given the polynomial equation $P(D) = D^2 + 2D + 6 = 0$ and asked to find its roots (zeros). 2. The roots can be found using the quadratic formula: $$D = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ where $a=1$, $b=2$, and $c=6$. 3. Calculate the discriminant: $$\Delta = b^2 - 4ac = 2^2 - 4 \times 1 \times 6 = 4 - 24 = -20$$ 4. Since $\Delta < 0$, the roots are complex. We proceed with: $$D = \frac{-2 \pm \sqrt{-20}}{2} = \frac{-2 \pm \sqrt{20}i}{2}$$ because $\sqrt{-1} = i$. 5. Simplify $\sqrt{20}$: $$\sqrt{20} = \sqrt{4 \times 5} = 2\sqrt{5}$$ 6. Substitute back to find roots: $$D = \frac{-2 \pm 2i\sqrt{5}}{2} = -1 \pm i\sqrt{5}$$ 7. Therefore, the zeros of the polynomial are $D_1 = -1 + i\sqrt{5}$ and $D_2 = -1 - i\sqrt{5}$. 8. These are complex conjugate roots, indicating the polynomial has no real zeros but two complex ones.