the greatest negative integer satisfying `x^2+4x-77<0` and `x^2>4` is

Options

A. `-4`
B. `-6`
C. `-7`
D. none of these

Answer

To solve the problem, we need to find the greatest negative integer satisfying the inequalities \( x^2 + 4x - 77 < 0 \) and \( x^2 > 4 \).

### Step 1: Solve the first inequality \( x^2 + 4x - 77 < 0 \)

1. **Factor the quadratic expression**: We can rewrite the inequality as:
\[ x^2 + 4x - 77 = (x - 7)(x + 11) < 0 \]
Here, we factored the quadratic by finding two numbers that multiply to -77 and add to 4, which are -7 and 11.

2. **Identify critical points**: The critical points are \( x = 7 \) and \( x = -11 \).

3. **Test intervals**: We will test the intervals determined by these critical points:
- For \( x < -11 \) (e.g., \( x = -12 \)):
\[ (-12 - 7)(-12 + 11) = (-19)(-1) > 0 \]
- For \( -11 < x < 7 \) (e.g., \( x = 0 \)):
\[ (0 - 7)(0 + 11) = (-7)(11) < 0 \]
- For \( x > 7 \) (e.g., \( x = 8 \)):
\[ (8 - 7)(8 + 11) = (1)(19) > 0 \]

4. **Conclusion for the first inequality**: The solution to the inequality \( x^2 + 4x - 77 < 0 \) is:
\[ -11 < x < 7 \]

### Step 2: Solve the second inequality \( x^2 > 4 \)

1. **Rewrite the inequality**: We can express this as:
\[ x^2 - 4 > 0 \]
This can be factored using the difference of squares:
\[ (x - 2)(x + 2) > 0 \]

2. **Identify critical points**: The critical points are \( x = -2 \) and \( x = 2 \).

3. **Test intervals**: We will test the intervals determined by these critical points:
- For \( x < -2 \) (e.g., \( x = -3 \)):
\[ (-3 - 2)(-3 + 2) = (-5)(-1) > 0 \]
- For \( -2 < x < 2 \) (e.g., \( x = 0 \)):
\[ (0 - 2)(0 + 2) = (-2)(2) < 0 \]
- For \( x > 2 \) (e.g., \( x = 3 \)):
\[ (3 - 2)(3 + 2) = (1)(5) > 0 \]

4. **Conclusion for the second inequality**: The solution to the inequality \( x^2 > 4 \) is:
\[ (-\infty, -2) \cup (2, \infty) \]

### Step 3: Find the intersection of the two solutions

1. **Combine the results**:
- From the first inequality: \( -11 < x < 7 \)
- From the second inequality: \( x < -2 \) or \( x > 2 \)

2. **Intersection**: The valid range from both inequalities is:
\[ -11 < x < -2 \]

### Step 4: Identify the greatest negative integer

1. The greatest negative integer within the interval \( -11 < x < -2 \) is \( -3 \).

### Final Answer

The greatest negative integer satisfying both inequalities is:
\[ \boxed{-3} \]