To determine the factors of the expression 3x²y² + 6x² + 12y² + 24 + 6x² + 2y² + 6x² + 4, we first simplify it. Let’s combine like terms:
1. Start with the expression: 3x²y² + 6x² + 12y² + 24 + 6x² + 2y² + 6x² + 4
2. Combine the terms:
- The x² terms: 6x² + 6x² + 6x² = 18x²
- The y² terms: 3x²y² + 12y² + 2y² = 3x²y² + 14y²
3. Now the expression simplifies to:
3x²y² + 18x² + 14y² + 24 + 4
Upon further inspection, we need to combine 24 + 4 = 28:
3x²y² + 18x² + 14y² + 28
Next, we can factor by grouping:
First, notice that 3 can be factored out from the entire expression:
3(x²y² + 6x² + 4y² + 9.33)
However, this doesn’t yield a simple factorization, so let’s look for another factorization approach.
Taking each term under consideration, observe:
- 3x²y² = 3(xy²)(x)
- 6x² = 6(x)(x)
- 12y² = 12(y)(y).
- The constant term 28 can be approached as 4(7).
Thus, one way to approach the complete factorization is:
(x + 2)(3y² + 7)(2x + 6)y + 4 (terms using the quadratic equation can also be factorized)
However, without further constraints, the full factorization requires the Solving Quadratic Formula. It is advisable to use a tool or software code for verification and to showcase the most reduced version.
In summary, the complete factorization is complex, but we arrive at potential factors of (3)(x²y² + 6x + 4y + 2).
For best practice, identifying the values of x and y that satisfy all terms will yield an optimized factoring scheme.