Solve: `sqrt(2+x-x^2) gt x-4`

Explanation

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

### Step 1: Square both sides
Since we have a square root on the left side, we will square both sides of the inequality to eliminate the square root. This gives us:
\[ 2 + x - x^2 > (x - 4)^2 \]

### Step 2: Expand the right side
Now, we will expand the right side:
\[ (x - 4)^2 = x^2 - 8x + 16 \]
So the inequality becomes:
\[ 2 + x - x^2 > x^2 - 8x + 16 \]

### Step 3: Rearrange the inequality
Next, we will move all terms to one side of the inequality:
\[ 2 + x - x^2 - x^2 + 8x - 16 > 0 \]
This simplifies to:
\[ -2x^2 + 9x - 14 > 0 \]
or multiplying through by -1 (which reverses the inequality):
\[ 2x^2 - 9x + 14 < 0 \]

### Step 4: Factor the quadratic
Now we need to factor the quadratic \( 2x^2 - 9x + 14 \). We can use the quadratic formula to find the roots:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} = \frac{9 \pm \sqrt{(-9)^2 - 4 \cdot 2 \cdot 14}}{2 \cdot 2} \]
Calculating the discriminant:
\[ b^2 - 4ac = 81 - 112 = -31 \]
Since the discriminant is negative, the quadratic \( 2x^2 - 9x + 14 \) does not cross the x-axis and is always positive.

### Step 5: Check the conditions for the square root
Since we squared both sides, we must ensure that the expression under the square root is non-negative:
\[ 2 + x - x^2 \geq 0 \]
Rearranging gives:
\[ -x^2 + x + 2 \geq 0 \]
or:
\[ x^2 - x - 2 \leq 0 \]
Factoring gives:
\[ (x - 2)(x + 1) \leq 0 \]

### Step 6: Find the intervals
The roots of the equation \( (x - 2)(x + 1) = 0 \) are \( x = 2 \) and \( x = -1 \). We can test intervals to find where the product is non-positive:
- For \( x < -1 \): both factors are negative, product is positive.
- For \( -1 < x < 2 \): one factor is negative, one is positive, product is negative.
- For \( x > 2 \): both factors are positive, product is positive.

Thus, the solution to \( (x - 2)(x + 1) \leq 0 \) is:
\[ -1 \leq x \leq 2 \]

### Final Solution
The solution to the original inequality \( \sqrt{2 + x - x^2} > x - 4 \) is:
\[ x \in [-1, 2] \]