To completely factor the expression 3x² + 21 + 3x² + 7 + 3x + 7x + 7 + 3x + 7x + 3, we first combine like terms and reorganize the expression.
1. **Combine Like Terms:**
- First, let’s combine all the x² terms:
- We have 3x² + 3x² = 6x².
- Next, we combine the x terms:
- 3x + 7x + 3x + 7x = 20x.
- Finally, we combine the constant numbers:
- 21 + 7 + 3 = 31.
2. **Reconstructed Expression:**
After combining all the like terms, we get:
6x² + 20x + 31
3. **Look for Common Factors:**
We can take out the greatest common factor:
- The common factor of 6 and 20 is 2.
This gives us:
2(3x² + 10x + 15.5)
4. **Factor the Quadratic Further:**
Now, we will complete the square or use the quadratic formula to factor 3x² + 10x + 15.5. This quadratic does not factor easily into integers, indicating this expression itself might be prime or require factoring using numerical methods.
5. **Conclude with Prime Factorization:**
Thus, the fully factored expression, as far as we can express in simpler terms without rational numbers, is:
2(3x² + 10x + 15.5)
For precise prime factored terms, additional methods or approximation may be necessary due to the decimal values in the quadratic equation.
In summary, the expression cannot be completely factored with whole numbers, and the most simplified result we achieve is:
2(3x² + 10x + 15.5)