%5E(3)%2B4(x%2By)%5E(2)%2B4x%2B4y.jpeg "(x+y)^(3)+4(x+y)^(2)+4x+4y")
Factoring the Expression (x+y)³ + 4(x+y)² + 4x + 4y
This expression can be factored using a combination of grouping and the sum of cubes pattern. Let's break down the steps:
1. Grouping Terms
First, let's group the terms with common factors:
(x+y)³ + 4(x+y)² + 4x + 4y = [(x+y)³ + 4(x+y)²] + [4x + 4y]
2. Factoring out Common Factors
Now, we factor out the common factors from each group:
[(x+y)³ + 4(x+y)²] + [4x + 4y] = (x+y)²(x+y+4) + 4(x+y)
3. Recognizing the Sum of Cubes Pattern
Notice that we now have a common factor of (x+y) in both terms. Let's factor that out:
(x+y)²(x+y+4) + 4(x+y) = (x+y)[(x+y)(x+y+4) + 4]
4. Simplifying the Expression
Let's simplify the expression inside the brackets:
(x+y)[(x+y)(x+y+4) + 4] = (x+y)[x² + 2xy + y² + 4x + 4y + 4]
5. Final Factored Form
Finally, we can rearrange the terms inside the brackets to get the fully factored form:
(x+y)[x² + 2xy + y² + 4x + 4y + 4] = (x+y)(x² + 2xy + y² + 4x + 4y + 4)
This is the factored form of the original expression (x+y)³ + 4(x+y)² + 4x + 4y.