The set of values of p for which the expression `x^(2)` −2px+3p+4 is negative for at least one real x is

Options

A. a. `p lt 1`
B. b. `p=0`
C. c. (−∞,−1) U (4,∞)
D. d. `p in (-(1)/(2),(1)/(2))`

Answer

To solve the problem, we need to determine the set of values of \( p \) for which the expression \( x^2 - 2px + 3p + 4 \) is negative for at least one real \( x \). This can be achieved by analyzing the discriminant of the quadratic expression.

### Step-by-Step Solution:

1. **Identify the Quadratic Coefficients**:
The given expression is \( x^2 - 2px + (3p + 4) \).
Here, we identify:
- \( a = 1 \)
- \( b = -2p \)
- \( c = 3p + 4 \)

2. **Determine the Condition for Negativity**:
For the quadratic expression to be negative for at least one real \( x \), the discriminant \( D \) must be greater than 0:
\[ D = b^2 - 4ac > 0 \]

3. **Calculate the Discriminant**:
Substitute the values of \( a \), \( b \), and \( c \) into the discriminant formula:
\[ D = (-2p)^2 - 4 \cdot 1 \cdot (3p + 4) \]
Simplifying this gives:
\[ D = 4p^2 - 4(3p + 4) \]
\[ D = 4p^2 - 12p - 16 \]

4. **Set Up the Inequality**:
We need to solve the inequality:
\[ 4p^2 - 12p - 16 > 0 \]
Dividing the entire inequality by 4 simplifies it:
\[ p^2 - 3p - 4 > 0 \]

5. **Factor the Quadratic**:
We can factor the quadratic expression:
\[ (p - 4)(p + 1) > 0 \]

6. **Determine the Intervals**:
To find where the product is greater than zero, we analyze the critical points \( p = -1 \) and \( p = 4 \). We test the intervals:
- For \( p < -1 \): Choose \( p = -2 \) → \( (-2 - 4)(-2 + 1) = (-6)(-1) > 0 \) (True)
- For \( -1 < p < 4 \): Choose \( p = 0 \) → \( (0 - 4)(0 + 1) = (-4)(1) < 0 \) (False)
- For \( p > 4 \): Choose \( p = 5 \) → \( (5 - 4)(5 + 1) = (1)(6) > 0 \) (True)

7. **Conclusion**:
The solution to the inequality \( (p - 4)(p + 1) > 0 \) gives us the intervals:
\[ p \in (-\infty, -1) \cup (4, \infty) \]

### Final Answer:
The set of values of \( p \) for which the expression \( x^2 - 2px + 3p + 4 \) is negative for at least one real \( x \) is:
\[ (-\infty, -1) \cup (4, \infty) \]