Let A be the set of values of k for which 2 lies between the roots of the quadratic equation `x^(2)+(k+2)x-(k+3)=0`, then A is given by

Options

A. `(-oo, -5)`
B. `(5, oo)`
C. `(-oo, -5]`
D. `[5, oo)`

Answer

To determine the set of values of \( k \) for which \( 2 \) lies between the roots of the quadratic equation \( x^2 + (k+2)x - (k+3) = 0 \), we will follow these steps:

### Step 1: Identify the quadratic function
The given quadratic equation is:
\[ f(x) = x^2 + (k+2)x - (k+3) \]

### Step 2: Condition for \( 2 \) to lie between the roots
For \( 2 \) to lie between the roots of the quadratic equation, the value of the function at \( x = 2 \) must be negative:
\[ f(2) < 0 \]

### Step 3: Calculate \( f(2) \)
Substituting \( x = 2 \) into the function:
\[ f(2) = 2^2 + (k+2) \cdot 2 - (k+3) \]
Calculating this gives:
\[ f(2) = 4 + 2k + 4 - k - 3 \]
Simplifying:
\[ f(2) = (2k - k) + (4 + 4 - 3) = k + 5 \]

### Step 4: Set the inequality
Now, we set up the inequality:
\[ k + 5 < 0 \]
This simplifies to:
\[ k < -5 \]

### Step 5: Condition for real and distinct roots
Next, we need to ensure that the quadratic has real and distinct roots. This is determined by the discriminant being greater than zero:
\[ D = (k+2)^2 - 4 \cdot 1 \cdot (-(k+3)) > 0 \]
Calculating the discriminant:
\[ D = (k+2)^2 + 4(k+3) \]
Expanding this:
\[ D = k^2 + 4k + 4 + 4k + 12 = k^2 + 8k + 16 \]
This can be factored as:
\[ D = (k + 4)^2 \]
For the discriminant to be greater than zero:
\[ (k + 4)^2 > 0 \]
This implies:
\[ k + 4 \neq 0 \quad \Rightarrow \quad k \neq -4 \]

### Step 6: Combine the conditions
From the conditions we have:
1. \( k < -5 \)
2. \( k \neq -4 \)

Since \( k < -5 \) already excludes \( k = -4 \), the set of values for \( k \) is:
\[ A = (-\infty, -5) \]

### Final Answer
Thus, the set \( A \) is given by:
\[ A = (-\infty, -5) \]