## Factoring the Cubic Expression: x³ + 6x²y + 12xy² + 7y³

This article explores the factorization of the cubic expression **x³ + 6x²y + 12xy² + 7y³**. We will use a combination of pattern recognition and the sum of cubes formula to achieve this.

### Recognizing a Pattern

First, we notice that the expression exhibits a specific pattern:

**Descending powers of x:**The powers of x decrease from 3 to 0.**Ascending powers of y:**The powers of y increase from 0 to 3.**Coefficients:**The coefficients (1, 6, 12, 7) seem to follow a pattern, but it's not immediately clear what it is.

This pattern suggests that the expression might be a perfect cube, but we need to investigate further.

### Applying the Sum of Cubes Formula

The sum of cubes formula states:

**a³ + b³ = (a + b)(a² - ab + b²)**

To apply this formula, we need to find expressions for 'a' and 'b' that satisfy the given cubic expression.

Let's try:

**a = x:**x³ matches the first term of the expression.**b = 2y:**(2y)³ = 8y³, which is not directly present in our expression.

Since the coefficient of the y³ term is 7, not 8, we need to adjust our approach slightly. We can rewrite the expression as:

**x³ + 6x²y + 12xy² + 7y³ = (x³ + 6x²y + 12xy² + 8y³) - y³**

Now, we can apply the sum of cubes formula to the first part of the expression:

**x³ + 6x²y + 12xy² + 8y³ = (x + 2y)(x² - 2xy + 4y²)**

Therefore, the complete factorization is:

**(x + 2y)(x² - 2xy + 4y²) - y³**

While we've factored the expression, we can't further simplify the remaining terms using the sum of cubes formula. The final factored form is:

**(x + 2y)(x² - 2xy + 4y²) - y³**

### Conclusion

By recognizing patterns and applying the sum of cubes formula, we were able to factor the cubic expression x³ + 6x²y + 12xy² + 7y³. This process demonstrates how understanding mathematical properties and formulas can be applied to solve complex algebraic problems.