If `-x^(2)+3x+4gt0`, then which one of the following is correct?

Options

A. `x""in(-1,4)`
B. `x""in[-1,4]`
C. `x""in(oo,-1)uu(4,oo)`
D. `x""in(-oo,-1)uu(4,oo)`

Explanation

To solve the inequality \(-x^2 + 3x + 4 > 0\), we will follow these steps:

### Step 1: Rewrite the inequality
The given inequality is:
\[ -x^2 + 3x + 4 > 0 \]
To make it easier to solve, we can multiply the entire inequality by \(-1\). Remember that multiplying by a negative number reverses the inequality sign:
\[ x^2 - 3x - 4 < 0 \]

### Step 2: Factor the quadratic expression
Next, we need to factor the quadratic expression \(x^2 - 3x - 4\). We are looking for two numbers that multiply to \(-4\) (the constant term) and add to \(-3\) (the coefficient of \(x\)). The numbers \(-4\) and \(1\) satisfy these conditions:
\[ x^2 - 4x + x - 4 = (x - 4)(x + 1) \]
Thus, we can rewrite the inequality as:
\[ (x - 4)(x + 1) < 0 \]

### Step 3: Determine the critical points
Now, we find the critical points by setting each factor equal to zero:
\[ x - 4 = 0 \quad \Rightarrow \quad x = 4 \]
\[ x + 1 = 0 \quad \Rightarrow \quad x = -1 \]
So, the critical points are \(x = -1\) and \(x = 4\).

### Step 4: Test intervals
We will test the intervals defined by these critical points: \((-∞, -1)\), \((-1, 4)\), and \((4, ∞)\).

1. **Interval \((-∞, -1)\)**: Choose \(x = -2\)
\[ (-2 - 4)(-2 + 1) = (-6)(-1) = 6 > 0 \quad \text{(not in the solution)} \]

2. **Interval \((-1, 4)\)**: Choose \(x = 0\)
\[ (0 - 4)(0 + 1) = (-4)(1) = -4 < 0 \quad \text{(in the solution)} \]

3. **Interval \((4, ∞)\)**: Choose \(x = 5\)
\[ (5 - 4)(5 + 1) = (1)(6) = 6 > 0 \quad \text{(not in the solution)} \]

### Step 5: Write the solution
The solution to the inequality \(x^2 - 3x - 4 < 0\) is:
\[ x \in (-1, 4) \]

### Conclusion
Thus, the correct answer is that \(x\) belongs to the open interval \((-1, 4)\).

---