The value of b and c for which the identity `f(x+1) - f (x) = 8x+3 `is satisfied, where f `(x) = bx^2 + cx+d`, are

Options

A. b=2,c=1
B. `b= 4,c=-1 `
C. `b=-1,c=4`
D. none

Answer

To solve the problem, we need to find the values of \( b \) and \( c \) such that the identity \( f(x+1) - f(x) = 8x + 3 \) holds, where \( f(x) = bx^2 + cx + d \).

### Step-by-step Solution:

1. **Write down the function**:
\[ f(x) = bx^2 + cx + d \]

2. **Find \( f(x+1) \)**:
\[ f(x+1) = b(x+1)^2 + c(x+1) + d \]
Expanding \( (x+1)^2 \):
\[ f(x+1) = b(x^2 + 2x + 1) + c(x + 1) + d \]
This simplifies to:
\[ f(x+1) = bx^2 + 2bx + b + cx + c + d \]
Combining like terms:
\[ f(x+1) = bx^2 + (2b + c)x + (b + c + d) \]

3. **Calculate \( f(x+1) - f(x) \)**:
\[ f(x+1) - f(x) = \left[ bx^2 + (2b + c)x + (b + c + d) \right] - \left[ bx^2 + cx + d \right] \]
This simplifies to:
\[ f(x+1) - f(x) = (2b + c - c)x + (b + c + d - d) \]
Which further simplifies to:
\[ f(x+1) - f(x) = 2bx + (b + c) \]

4. **Set the equation equal to \( 8x + 3 \)**:
\[ 2bx + (b + c) = 8x + 3 \]

5. **Compare coefficients**:
- For the coefficient of \( x \):
\[ 2b = 8 \quad \Rightarrow \quad b = 4 \]
- For the constant term:
\[ b + c = 3 \]
Substituting \( b = 4 \) into the equation:
\[ 4 + c = 3 \quad \Rightarrow \quad c = 3 - 4 = -1 \]

6. **Final values**:
\[ b = 4, \quad c = -1 \]

### Conclusion:
The values of \( b \) and \( c \) for which the identity \( f(x+1) - f(x) = 8x + 3 \) is satisfied are:
\[ b = 4, \quad c = -1 \]