Understanding the Equation
The equation to solve is:
(3x + 1)(2x - 3)(x + 3)(3x - 1) + 16
We need to determine the value of x that satisfies this equation.
Expanding the Equation
1. Multiply the Factors:
- Start by multiplying the first two factors:
(3x + 1)(2x - 3) = 6x^2 - 9x + 2x - 3 = 6x^2 - 7x - 3.
2. Multiply the Remaining Factors:
- Next, multiply (x + 3) and (3x - 1):
(x + 3)(3x - 1) = 3x^2 - x + 9x - 3 = 3x^2 + 8x - 3.
3. Combine the Results:
- Now multiply the two products obtained:
(6x^2 - 7x - 3)(3x^2 + 8x - 3).
Setting Up the Equation
4. Setting Equal to Zero:
- Set the entire expression equal to zero:
(6x^2 - 7x - 3)(3x^2 + 8x - 3) + 16 = 0.
5. Isolate the Polynomial:
- Rearranging yields a polynomial equation that can be solved for x.
Finding the Roots
6. Using Possible Values:
- Substitute the potential values from the options (7, 8, 14, 22, 43, 34, 5) into the polynomial.
- Check which value satisfies the equation.
7. Confirm the Solution:
- Evaluate each option until you find a value that makes the left side equal to zero.
Conclusion
Through this process, the value of x can be determined. The solution involves algebraic manipulation and testing possible values to find the correct answer.
By systematically applying these steps, you will arrive at the solution efficiently.