Final answer:
The factored form of 4x²+25y² over the complex numbers is (2x+5iy)(2x-5iy), using the imaginary unit i, where i²=-1. This enables the expression to factor as a difference of squares.
Step-by-step explanation:
The factored form of the expression 4x²+25y² over the complex numbers is not factorable using real numbers because it is a sum of squares.
However, over the complex numbers, we can use the imaginary unit i where i² = -1.
We can express 4x² as (2x)² and 25y² as (5y)².
Because the imaginary unit squared gives us -1, we can write 25y² as -(-25y²), which is -(5iy)².
This allows the expression to be written as a difference of squares: 4x² - (5iy)², which factors as (2x+5iy) and (2x-5iy).
Therefore, the correct factored form of the expression over the complex numbers is (2x+5iy)(2x-5iy), which is option C).