Let `P(x)` be quadratic polynomical with real coefficient such tht for all real `x(1)` the relation `2(1 + P(x)) = P(x - 1) + P(x + 1)` holds. If `P(0) = 8` and `P(2) = 32` then
If the range of `P(x)` is `[m, oo)` then `'m'` is less then

Options

A. `-12`
B. 15
C. `-17`
D. `-5`

Answer

To solve the problem step by step, we will start with the given conditions and work through the quadratic polynomial.

### Step 1: Define the Quadratic Polynomial
Let \( P(x) = ax^2 + bx + c \).

### Step 2: Use the Given Condition \( P(0) = 8 \)
Substituting \( x = 0 \):
\[ P(0) = a(0)^2 + b(0) + c = c = 8 \]
Thus, we have:
\[ c = 8 \]

### Step 3: Use the Given Condition \( P(2) = 32 \)
Substituting \( x = 2 \):
\[ P(2) = a(2)^2 + b(2) + c = 32 \]
Substituting \( c = 8 \):
\[ 4a + 2b + 8 = 32 \]
Simplifying this:
\[ 4a + 2b = 32 - 8 = 24 \]
Dividing the entire equation by 2:
\[ 2a + b = 12 \quad \text{(Equation 1)} \]

### Step 4: Use the Given Relation
The relation given is:
\[ 2(1 + P(x)) = P(x - 1) + P(x + 1) \]
Expanding both sides:
- Left side:
\[ 2(1 + P(x)) = 2 + 2P(x) = 2 + 2(ax^2 + bx + 8) = 2 + 2ax^2 + 2bx + 16 = 2ax^2 + 2bx + 18 \]

- Right side:
\[ P(x - 1) = a(x - 1)^2 + b(x - 1) + 8 = a(x^2 - 2x + 1) + b(x - 1) + 8 \]
\[ = ax^2 - 2ax + a + bx - b + 8 = ax^2 + (-2a + b)x + (a - b + 8) \]
\[ P(x + 1) = a(x + 1)^2 + b(x + 1) + 8 = a(x^2 + 2x + 1) + b(x + 1) + 8 \]
\[ = ax^2 + 2ax + a + bx + b + 8 = ax^2 + (2a + b)x + (a + b + 8) \]
Adding these:
\[ P(x - 1) + P(x + 1) = [ax^2 + (-2a + b)x + (a - b + 8)] + [ax^2 + (2a + b)x + (a + b + 8)] \]
\[ = 2ax^2 + (0)x + (2a + 16) \]

### Step 5: Set the Two Sides Equal
Equating both sides:
\[ 2ax^2 + 18 = 2ax^2 + 2a + 16 \]
This simplifies to:
\[ 18 = 2a + 16 \]
Solving for \( a \):
\[ 2a = 18 - 16 = 2 \implies a = 1 \]

### Step 6: Substitute \( a \) Back to Find \( b \)
Using Equation 1:
\[ 2(1) + b = 12 \implies 2 + b = 12 \implies b = 10 \]

### Step 7: Write the Polynomial
Now we have:
\[ P(x) = x^2 + 10x + 8 \]

### Step 8: Find the Range of \( P(x) \)
To find the range of the quadratic polynomial, we can complete the square:
\[ P(x) = (x^2 + 10x + 25) - 25 + 8 = (x + 5)^2 - 17 \]
The vertex of this parabola is at \( (-5, -17) \). Since the parabola opens upwards, the minimum value of \( P(x) \) is \( -17 \).

### Conclusion
The range of \( P(x) \) is \( [-17, \infty) \). Therefore, \( m = -17 \).

### Final Answer
Since the question asks for \( m \) to be less than a certain value, we conclude:
\[ m < -5 \]