Which expression represents 625p4-16

when factored completely over the complex numbers?

(5p-2i) 2 (5p+2i) 2

(25p2+4i) (25p2-4i)

(25p2+4i) (5p-2i) (5p+2i)

(5p-2) (5p+2) (5p-2i) (5p+2i)

Answer: the fourth option (5p - 2) (5p + 2) (5p - 2i) (5p + 2i)

Explanation:

1) Express the given expression as the difference of two squares:

625p⁴ - 16 = (25p²) ² - 4²

2) Factor as the product of a sum times its difference:

(25p²) ² - 4² = (25p² - 4) (25p² + 4)

3) Factor each binomial (again square of a difference and sum times its difference):

25p² - 4 = (5p) ² - 2² = (5p - 2) (5p + 2)

25p² + 4 = (5p) ² - (2i) ² = (5p - 2i) (5p + 2i)

4) Therefore, the 4 factors are:

(5p - 2) (5p + 2) (5p - 2i) (5p + 2i)