Factoring and Expanding the Expression (x⁴ + 1)(x² + 1)(x + 1)(x  1)
This expression presents a fascinating opportunity to explore the interplay between factorization and expansion in algebra. Let's break down the process stepbystep.
Recognizing Key Patterns
 Difference of Squares: Notice the (x + 1) and (x  1) terms. These represent the classic difference of squares pattern: (a + b)(a  b) = a²  b²
 Sum of Squares: The (x⁴ + 1) and (x² + 1) terms might seem less familiar, but they are both irreducible over the real numbers.
StepbyStep Solution

Initial Expansion: Begin by expanding the difference of squares: (x + 1)(x  1) = x²  1

Substitution: Substitute the result into the original expression: (x⁴ + 1)(x² + 1)(x²  1)

Further Expansion: Now, we can expand the product of the remaining terms: (x⁴ + 1)(x⁴  1)

Final Difference of Squares: We encounter another difference of squares: (x⁸  1)
The Final Result
The fully expanded form of the expression is x⁸  1.
Key Takeaways
 Strategic Factoring: Identifying patterns like the difference of squares allows for efficient simplification.
 Irreducible Factors: Some expressions, like (x⁴ + 1) and (x² + 1), may not be factorable over the real numbers.
 Expansion and Factorization: These two operations are often used in tandem to manipulate expressions and solve problems in algebra.