Determine the value of k for which `k^2+4k + 8, 2k^2+ 3k + 6 and 3k^2 +4k + 4` are in A.P.

Answer

To determine the value of \( k \) for which the expressions \( k^2 + 4k + 8 \), \( 2k^2 + 3k + 6 \), and \( 3k^2 + 4k + 4 \) are in Arithmetic Progression (A.P.), we can follow these steps:

### Step 1: Understand the condition for A.P.
For three terms \( A \), \( B \), and \( C \) to be in A.P., the following condition must hold:
\[ 2B = A + C \]

### Step 2: Assign the terms
Let:
- \( A = k^2 + 4k + 8 \)
- \( B = 2k^2 + 3k + 6 \)
- \( C = 3k^2 + 4k + 4 \)

### Step 3: Apply the A.P. condition
According to the A.P. condition:
\[ 2(2k^2 + 3k + 6) = (k^2 + 4k + 8) + (3k^2 + 4k + 4) \]

### Step 4: Simplify the left side
Calculating the left side:
\[ 2(2k^2 + 3k + 6) = 4k^2 + 6k + 12 \]

### Step 5: Simplify the right side
Calculating the right side:
\[ (k^2 + 4k + 8) + (3k^2 + 4k + 4) = k^2 + 3k^2 + 4k + 4k + 8 + 4 = 4k^2 + 8k + 12 \]

### Step 6: Set the two sides equal
Now we set the left side equal to the right side:
\[ 4k^2 + 6k + 12 = 4k^2 + 8k + 12 \]

### Step 7: Simplify the equation
Subtract \( 4k^2 + 12 \) from both sides:
\[ 6k = 8k \]

### Step 8: Rearrange the equation
Rearranging gives:
\[ 0 = 8k - 6k \]
\[ 0 = 2k \]

### Step 9: Solve for \( k \)
Dividing both sides by 2:
\[ k = 0 \]

### Final Answer
Thus, the value of \( k \) for which the given expressions are in A.P. is:
\[ \boxed{0} \]