The expression \( 16 + x^4 \) can be factored using the sum of squares identity. Here's how it works:

\[
16 + x^4 = (x^2)^2 + 4^2
\]

This can be rewritten as:

\[
16 + x^4 = (x^2 + 2i)(x^2 - 2i)
\]

Thus, the factorization of \( 16 + x^4 \) is:

\[
(x^2 + 2i)(x^2 - 2i)
\]

**Note:** The factorization involves imaginary numbers, where \( i \) is the imaginary unit with \( i^2 = -1 \).

Would you like more details, or do you have any other questions?

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Here are some related questions you might find useful:

1. How do you factor expressions using the difference of squares?
2. What is the significance of imaginary numbers in algebra?
3. Can all quadratic expressions be factored into real numbers?
4. How can you factor higher-degree polynomials?
5. What are some common factoring techniques for polynomials?
6. What are the roots of the equation \(16 + x^4 = 0\)?
7. How does the sum of squares formula differ from the difference of squares?
8. Can you explain how to solve equations involving complex numbers?

**Tip:** Remember that not all polynomial expressions can be factored into real numbers; some require complex or imaginary numbers.

Explore the factorization of the expression 16 + x^4 using the sum of squares identity and imaginary numbers. This problem delves into polynomial factorization techniques and complex numbers, suitable for advanced high school students.

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