## Factoring the Expression (x+y)³ + 4(x+y)² + 4x + 4y

This article explores the process of factoring the expression (x+y)³ + 4(x+y)² + 4x + 4y. We will utilize techniques of **substitution** and **factoring by grouping** to simplify the expression and achieve its factored form.

### Step 1: Substitution

To simplify the expression, we can use substitution. Let's replace (x+y) with a single variable, say 'a'. This gives us:

**(x+y)³ + 4(x+y)² + 4x + 4y = a³ + 4a² + 4x + 4y**

### Step 2: Factoring by Grouping

Now, we can factor the expression by grouping. We can rewrite the expression as:

**a³ + 4a² + 4x + 4y = (a³ + 4a²) + (4x + 4y)**

Next, we factor out the common factors from each group:

**(a³ + 4a²) + (4x + 4y) = a²(a+4) + 4(x+y)**

### Step 3: Substituting Back

Finally, we substitute back (x+y) for 'a':

**a²(a+4) + 4(x+y) = (x+y)²(x+y+4) + 4(x+y)**

### Step 4: Final Factorization

Now we have the expression in a simplified factored form. We can further factor out (x+y) to get:

**(x+y)²(x+y+4) + 4(x+y) = (x+y)[(x+y)(x+y+4) + 4]**

### Conclusion

Therefore, the factored form of the expression (x+y)³ + 4(x+y)² + 4x + 4y is **(x+y)[(x+y)(x+y+4) + 4]**.

By using substitution and factoring by grouping, we were able to successfully simplify the expression and reveal its hidden factors. This approach can be applied to various expressions involving similar patterns, making it a valuable technique for algebraic manipulation.