What is the complete factorization of the polynomial function over the set of complex numbers? f(x)=x3+3x2+16x+48 enter your answer in the box.

(x + 3) (x + 4i) (x - 4i)

Explanation:

1) Given: x³ + 3x² + 16x + 48

2) Group terms: (x³ + 3x²) + (16x + 48)

3) Common factors: x² (first group) and 16 (second group):

4) Common factor x + 3: (x + 3) (x² + 16)

5) Trick: i² = -1 ⇒ - i² = 1 ⇒ 16 = - 16i² = - (4i)²

   ⇒ (x + 3) [ x² - (4i)² ]

6) Factoring difference of squares: a² - b² = (a + b)(a - b)

⇒ (x + 3) [ x² - (4i)² ] = (x + 3) ( x + 4i ) (x - 4i )

Result:  (x + 3) ( x + 4i ) (x - 4i ) ← complete factorization

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