The comlete set of values of m for which the inequality `(x ^(2) -mx-2)/( x ^(2) + mx +4)gt-1` is satisfied `aa x in R,` is :

Options

A. `m =0`
B. ` -1 lt m lt 1`
C. `-2 lt m lt 2`
D. `-4 lt m lt 4`

Answer

To solve the inequality \(\frac{x^2 - mx - 2}{x^2 + mx + 4} > -1\), we will follow these steps:

### Step 1: Rearranging the Inequality
We start with the given inequality:
\[ \frac{x^2 - mx - 2}{x^2 + mx + 4} > -1 \]
We can rearrange this by moving \(-1\) to the left side:
\[ \frac{x^2 - mx - 2}{x^2 + mx + 4} + 1 > 0 \]

### Step 2: Combining the Fractions
Next, we combine the fractions:
\[ \frac{x^2 - mx - 2 + (x^2 + mx + 4)}{x^2 + mx + 4} > 0 \]
This simplifies to:
\[ \frac{2x^2 + 2}{x^2 + mx + 4} > 0 \]

### Step 3: Factoring the Numerator
We can factor out a 2 from the numerator:
\[ \frac{2(x^2 + 1)}{x^2 + mx + 4} > 0 \]
Since \(2\) is always positive, we can ignore it for the inequality:
\[ \frac{x^2 + 1}{x^2 + mx + 4} > 0 \]

### Step 4: Analyzing the Numerator
The numerator \(x^2 + 1\) is always positive for all \(x \in \mathbb{R}\) because \(x^2 \geq 0\).

### Step 5: Analyzing the Denominator
Now we need to ensure that the denominator \(x^2 + mx + 4\) is always positive. For this to happen, the discriminant of the quadratic must be less than zero:
\[ D = m^2 - 4ac = m^2 - 4 \cdot 1 \cdot 4 < 0 \]
This simplifies to:
\[ m^2 - 16 < 0 \]

### Step 6: Solving the Discriminant Inequality
Factoring the inequality:
\[ (m - 4)(m + 4) < 0 \]
This inequality holds true when \(m\) is between the roots:
\[ -4 < m < 4 \]

### Final Answer
Thus, the complete set of values of \(m\) for which the inequality is satisfied is:
\[ m \in (-4, 4) \]