Exploring the Expansion of (a + b)²(a²  ab + b²)²
This article will delve into the expansion of the algebraic expression (a + b)²(a²  ab + b²)². We'll explore the various methods for expanding this expression, ultimately aiming for a simplified and concise representation.
Understanding the Components
Before diving into the expansion, let's break down the expression into its constituent parts:
 (a + b)²: This is a perfect square trinomial, representing the square of the binomial (a + b).
 (a²  ab + b²)²: This represents the square of a trinomial, which can be viewed as a sum of cubes with a slight adjustment.
Expansion Methods
There are a couple of approaches we can use to expand this expression:
1. Direct Multiplication:
This method involves directly multiplying the terms within each set of parentheses.
 Step 1: Expand (a + b)² = a² + 2ab + b²
 Step 2: Expand (a²  ab + b²)² = (a²  ab + b²)(a²  ab + b²)
 This requires multiplying each term in the first trinomial with every term in the second trinomial.
 Step 3: Multiply the expanded forms of (a + b)² and (a²  ab + b²)².
This method, while straightforward, can lead to lengthy calculations and require careful attention to sign management.
2. Using Identities:
We can leverage algebraic identities to simplify the expansion:

Identity 1: (a + b)² = a² + 2ab + b²

Identity 2: (a  b)² = a²  2ab + b²

Identity 3: a³ + b³ = (a + b)(a²  ab + b²)

Step 1: Expand (a + b)² using Identity 1.

Step 2: Recognize that (a²  ab + b²)² is the square of the expression in Identity 3.

Step 3: Apply Identity 3 to simplify (a²  ab + b²)².
Using identities can significantly reduce the amount of calculation required.
Simplifying the Result
Once expanded, the expression will likely contain multiple terms. To simplify the result, combine like terms and look for any patterns or further factorization opportunities.
Example
Let's demonstrate the expansion using the identity method:
 Expand (a + b)²: (a + b)² = a² + 2ab + b²
 Apply Identity 3 to (a²  ab + b²)²: (a²  ab + b²)² = [(a + b)(a²  ab + b²)]² = (a³ + b³)²
 Expand (a³ + b³)²: (a³ + b³)² = a⁶ + 2a³b³ + b⁶
Therefore, the expanded form of (a + b)²(a²  ab + b²)² is a⁶ + 2a³b³ + b⁶.
Conclusion
Expanding (a + b)²(a²  ab + b²)² can be achieved using direct multiplication or leveraging algebraic identities. While direct multiplication is more straightforward, utilizing identities often leads to a more efficient and concise solution. Remember to carefully manage signs and combine like terms when simplifying the expanded expression.