Answer:
f(x) = x³ + 6x² + 13x + 10
Step-by-step explanation:
Note that complex roots occur in conjugate pairs, thus
x = - 2 + i is a root then x = - 2 - i is also a root
The roots are x = - 2, x = - 2 + i, x = - 2 - i, then the factors are
(x + 2), (x - (- 2 + i)) , (x - (- 2 - i)), that is
(x + 2), (x + 2 - i)(x + 2 + i)
The polynomial is the product of the factors
f(x) = (x + 2)(x + 2 - i)(x + 2 + i) ← expanding
= (x + 2)(x² + 2x + xi + 2x + 4 + 2i - xi - 2i - i²) → i² = - 1
= (x + 2)(x² + 4x + 4 + 1)
= (x + 2)(x² + 4x + 5) ← distribute
= x³ + 4x² + 5x + 2x² + 8x + 10 ← collect like terms
= x³ + 6x² + 13x + 10
