Final answer:
The expression x² + 20 is factored over the complex numbers as (x + 2√5i)(x - 2√5i), which is represented by Option A.
Step-by-step explanation:
To factor the expression x² + 20 over the complex numbers, we look for two complex numbers whose product is x² + 20. This involves finding two conjugates that, when multiplied together, will result in the given expression. A pair of complex conjugates has the form (a + bi)(a - bi), which equals a² + b² because the i² term (equal to -1) will cancel out the middle terms.
Given that the expression is x² + 20, we are essentially looking for a complex number a + bi such that (a + bi)(a - bi) = a² + b² = x² + 20. Here, a is x, and b² must be 20. Since b² is 20, b must be √20. Simplifying, we get b = √(4√5)√5, which equals 2√5. Therefore, the correct factorization is (x + 2√5i)(x - 2√5i).
The correct option that represents this factorization is Option A: (x + 2√5i)(x - 2√5i). Consequently, the please mention correct option in final answer is Option A.