1. **State the problem:** Factor the polynomial completely: $$8x^2 + 50$$. 2. **Identify common factors:** First, look for the greatest common factor (GCF) of the terms 8x^2 and 50. 3. The GCF of 8 and 50 is 2, so factor out 2: $$8x^2 + 50 = 2(4x^2 + 25)$$ 4. **Check if the remaining polynomial can be factored further:** The expression inside the parentheses is $$4x^2 + 25$$. 5. Recognize that $$4x^2 + 25$$ is a sum of squares since $$4x^2 = (2x)^2$$ and $$25 = 5^2$$. 6. The sum of squares does not factor over the real numbers but can be factored over the complex numbers as: $$a^2 + b^2 = (a + bi)(a - bi)$$ 7. Applying this to $$4x^2 + 25$$: $$4x^2 + 25 = (2x + 5i)(2x - 5i)$$ 8. **Write the complete factorization:** $$8x^2 + 50 = 2(2x + 5i)(2x - 5i)$$ 9. **Answer:** The correct factorization is option D: $$2 (2x + 5i) (2x - 5i)$$. This shows the polynomial factored completely over the complex numbers.