To completely factor the expression 49x² + 81 + 7x + 92 + 7x + 92 + 7x + 97x + 9 + 7x + 9, we first need to simplify it.
1. **Combine like terms**: The expression has several x terms and constant terms that can be combined.
- Combine the x terms:
- 7x + 7x + 7x + 97x + 7x = 7x(5 + 97) = 99x.
- Combine the constant terms:
- 81 + 92 + 92 + 9 + 9 = 281.
So, our simplified expression looks like this:
49x² + 99x + 281
2. **Look for a factoring method**: We can use the method of splitting the middle term or the quadratic formula if needed. In this case, we will check for factoring.
3. **Factoring the expression**: We are looking for two numbers that multiply to (49 * 281) and add up to 99. However, finding these numbers directly can be complicated due to the size of these numbers.
4. **Using the quadratic formula**: If we aren’t able to factor easily, we can apply the quadratic formula:
x = rac{-b
bsp; ext{±}
bsp; ext{sqrt}(b² – 4ac)}{2a}
Here, a = 49, b = 99, and c = 281:
- b² – 4ac = 99² – 4(49)(281)
- = 9801 – 55116
- = -45315 (a negative discriminant implies the original quadratic cannot be factored over the reals).
Since the discriminant is negative, the expression 49x² + 99x + 281 does not factor neatly over the reals.
Thus, the expression is not factorable in a simple way; however, it can be expressed in its standard form:
Final output: 49x² + 99x + 281
In conclusion, this expression is primarily expressed as a quadratic polynomial and does not factor completely with real-number solutions.