`-x^2+2x+8ge0`

Options

A. `(-2,6)
B. `(2,-4)`
C. `[-2,4]
D. `(-oo,-2]uu(-2,4)uu[4,oo)`

Answer

To solve the inequality \(-x^2 + 2x + 8 \geq 0\), we can follow these steps:

### Step 1: Rearrange the Inequality
Start by moving all terms to one side of the inequality:

\[ -x^2 + 2x + 8 \geq 0 \]

### Step 2: Factor Out the Negative Sign
Factor out a negative sign from the left side:

\[ -(x^2 - 2x - 8) \geq 0 \]

This can be rewritten as:

\[ x^2 - 2x - 8 \leq 0 \]

### Step 3: Factor the Quadratic
Now, we need to factor the quadratic expression \(x^2 - 2x - 8\). We look for two numbers that multiply to \(-8\) and add to \(-2\). The numbers \(-4\) and \(2\) work:

\[ x^2 - 2x - 8 = (x - 4)(x + 2) \]

Thus, we rewrite the inequality:

\[ (x - 4)(x + 2) \leq 0 \]

### Step 4: Find the Critical Points
Set each factor equal to zero to find the critical points:

1. \(x - 4 = 0 \Rightarrow x = 4\)
2. \(x + 2 = 0 \Rightarrow x = -2\)

### Step 5: Test Intervals
Now we will test the intervals determined by the critical points \(-2\) and \(4\):

1. **Interval 1**: \( (-\infty, -2) \)
- Choose \(x = -3\): \(((-3) - 4)((-3) + 2) = (-7)(-1) = 7 > 0\)

2. **Interval 2**: \( (-2, 4) \)
- Choose \(x = 0\): \((0 - 4)(0 + 2) = (-4)(2) = -8 < 0\)

3. **Interval 3**: \( (4, \infty) \)
- Choose \(x = 5\): \((5 - 4)(5 + 2) = (1)(7) = 7 > 0\)

### Step 6: Determine the Solution Set
We need the intervals where the product is less than or equal to zero. From our tests:

- The product is negative in the interval \((-2, 4)\).
- At the critical points \(x = -2\) and \(x = 4\), the product equals zero.

Thus, the solution set is:

\[ [-2, 4] \]

### Final Solution
The solution to the inequality \(-x^2 + 2x + 8 \geq 0\) is:

\[ [-2, 4] \]