Solve, `(1-sqrt(21-4x-x^(2)))/(x+1) ge 0`.

Explanation

To solve the inequality \((1 - \sqrt{21 - 4x - x^2})/(x + 1) \geq 0\), we will follow these steps:

### Step 1: Identify the conditions for the square root
The expression inside the square root must be non-negative:
\[ 21 - 4x - x^2 \geq 0 \]
This can be rearranged to:
\[ -x^2 - 4x + 21 \geq 0 \]
Multiplying through by -1 (which reverses the inequality):
\[ x^2 + 4x - 21 \leq 0 \]

### Step 2: Factor the quadratic
Next, we need to factor the quadratic \(x^2 + 4x - 21\):
\[ x^2 + 4x - 21 = (x - 3)(x + 7) \]
Thus, we have:
\[ (x - 3)(x + 7) \leq 0 \]

### Step 3: Find the critical points
The critical points from the factors are \(x = 3\) and \(x = -7\). We will test intervals around these points to determine where the product is non-positive.

### Step 4: Test intervals
We will test the intervals:
1. \( (-\infty, -7) \)
2. \( (-7, 3) \)
3. \( (3, \infty) \)

- For \(x < -7\) (e.g., \(x = -8\)):
\((x - 3)(x + 7) = (-8 - 3)(-8 + 7) = (-11)(-1) > 0\)

- For \(-7 < x < 3\) (e.g., \(x = 0\)):
\((x - 3)(x + 7) = (0 - 3)(0 + 7) = (-3)(7) < 0\)

- For \(x > 3\) (e.g., \(x = 4\)):
\((x - 3)(x + 7) = (4 - 3)(4 + 7) = (1)(11) > 0\)

Thus, the solution to \((x - 3)(x + 7) \leq 0\) is:
\[ -7 \leq x \leq 3 \]

### Step 5: Consider the denominator
Next, we must ensure that the denominator \(x + 1\) is not zero:
\[ x + 1 \neq 0 \implies x \neq -1 \]

### Step 6: Combine the results
From the previous steps, we have:
1. The solution from the square root condition: \([-7, 3]\)
2. The condition from the denominator: \(x \neq -1\)

Thus, the final solution is:
\[ [-7, -1) \cup (-1, 3] \]

### Final Answer
The solution to the inequality \((1 - \sqrt{21 - 4x - x^2})/(x + 1) \geq 0\) is:
\[ [-7, -1) \cup (-1, 3] \]