If `4 lt x lt 8` then the value of `12x -x^(2) - 32` is

Options

A. Zero
B. Positive
C. Negative
D. Not determinable

Answer

To solve the problem, we need to analyze the expression \( 12x - x^2 - 32 \) given the constraint \( 4 < x < 8 \).

### Step-by-Step Solution:

1. **Rewrite the Expression**:
We start with the expression:
\[ 12x - x^2 - 32 \]
Rearranging it gives:
\[ -x^2 + 12x - 32 \]

2. **Factor the Expression**:
To factor the quadratic expression, we can first multiply through by -1 to make it easier to factor:
\[ x^2 - 12x + 32 \]
Now, we need to factor this quadratic. We look for two numbers that multiply to \( 32 \) and add to \( -12 \). The numbers are \( -4 \) and \( -8 \):
\[ (x - 4)(x - 8) \]
Therefore, we can rewrite the original expression as:
\[ -(x - 4)(x - 8) \]

3. **Analyze the Factors**:
The expression \( -(x - 4)(x - 8) \) is a downward-opening parabola (since the leading coefficient is negative). The roots of the equation are \( x = 4 \) and \( x = 8 \).

4. **Determine the Sign of the Expression**:
We need to evaluate the sign of the expression in the interval \( 4 < x < 8 \).
- For \( x \) values between \( 4 \) and \( 8 \), both factors \( (x - 4) \) and \( (x - 8) \) will have the following signs:
- \( (x - 4) > 0 \) (positive)
- \( (x - 8) < 0 \) (negative)
- Therefore, the product \( (x - 4)(x - 8) < 0 \) in this interval.

5. **Conclusion**:
Since the expression is negative due to the negative sign in front, we conclude that:
\[ 12x - x^2 - 32 < 0 \quad \text{for} \quad 4 < x < 8 \]

### Final Answer:
The value of \( 12x - x^2 - 32 \) is negative for \( 4 < x < 8 \).