Factoring and Expanding (x+2)(x2)(x+3)(x3)
This expression represents the product of four binomials: (x+2), (x2), (x+3), and (x3). Let's explore how to factor and expand this expression.
Understanding the Pattern
Notice that each binomial is in the form of (x + a)(x  a), which is a difference of squares pattern.
Key point: The difference of squares pattern states that (x + a)(x  a) = x²  a².
Factoring the Expression

Factor the first two binomials: (x + 2)(x  2) = x²  2² = x²  4

Factor the last two binomials: (x + 3)(x  3) = x²  3² = x²  9

Combine the factored expressions: (x²  4)(x²  9)

Apply the difference of squares pattern again: (x²  4)(x²  9) = (x²  2²)(x²  3²) = (x + 2)(x  2)(x + 3)(x  3)
Expanding the Expression

Expand the first two factors: (x²  4)(x²  9) = x⁴  9x²  4x² + 36

Combine like terms: x⁴  9x²  4x² + 36 = x⁴  13x² + 36
Therefore, the expanded form of (x+2)(x2)(x+3)(x3) is x⁴  13x² + 36.
Conclusion
By recognizing and applying the difference of squares pattern, we effectively factored and expanded the given expression. This highlights the importance of identifying patterns in mathematics for simplifying complex expressions.