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 solve the problem, we need to determine the set of values of \( k \) for which the number 2 lies between the roots of the quadratic equation:

\[ x^2 + (k + 2)x - (k + 3) = 0 \]

### Step 1: Identify the quadratic equation
The quadratic equation can be expressed in the standard form \( ax^2 + bx + c = 0 \) where:
- \( a = 1 \)
- \( b = k + 2 \)
- \( c = -(k + 3) \)

### Step 2: Use the condition for roots
For 2 to lie between the roots of the quadratic equation, the value of the quadratic at \( x = 2 \) must be negative. Thus, we need to evaluate \( f(2) \):

\[ f(2) = 2^2 + (k + 2) \cdot 2 - (k + 3) \]

### Step 3: Simplify \( f(2) \)
Calculating \( f(2) \):

\[ f(2) = 4 + 2(k + 2) - (k + 3) \]

Expanding this:

\[ f(2) = 4 + 2k + 4 - k - 3 \]

Combining like terms:

\[ f(2) = (2k - k) + (4 + 4 - 3) = k + 5 \]

### Step 4: Set the inequality
For 2 to lie between the roots, we require:

\[ f(2) < 0 \]

Thus, we set up the inequality:

\[ k + 5 < 0 \]

### Step 5: Solve the inequality
To find the values of \( k \):

\[ k < -5 \]

### Conclusion
The set \( A \) of values of \( k \) for which 2 lies between the roots of the quadratic equation is:

\[ A = (-\infty, -5) \]