If: `f(x)=bx^2+cx+d and: f(x+1)-f(x)=8x+3,` then:

Options

A. `b=2,c=1`
B. `b=4,c=-1`
C. `b=-1,c=4`
D. `b=-1,c=1`

Answer

To solve the problem step by step, we start with the given function and the equation provided.

### Given:
1. \( f(x) = bx^2 + cx + d \)
2. \( f(x+1) - f(x) = 8x + 3 \)

### Step 1: Calculate \( f(x+1) \)
We need to find \( f(x+1) \):
\[ f(x+1) = b(x+1)^2 + c(x+1) + d \]
Expanding \( (x+1)^2 \):
\[ = b(x^2 + 2x + 1) + c(x + 1) + d \]
\[ = bx^2 + 2bx + b + cx + c + d \]
Combining like terms:
\[ = bx^2 + (2b + c)x + (b + c + d) \]

### Step 2: Calculate \( f(x+1) - f(x) \)
Now, we find \( f(x+1) - f(x) \):
\[ f(x+1) - f(x) = [bx^2 + (2b + c)x + (b + c + d)] - [bx^2 + cx + d] \]
Cancelling out \( bx^2 \) and \( d \):
\[ = (2b + c - c)x + (b + c + d - d) \]
\[ = 2bx + (b + c) \]

### Step 3: Set the equation equal to \( 8x + 3 \)
Now we set the expression equal to \( 8x + 3 \):
\[ 2bx + (b + c) = 8x + 3 \]

### Step 4: Compare coefficients
From the equation \( 2bx + (b + c) = 8x + 3 \), we can compare coefficients:
1. Coefficient of \( x \):
\[ 2b = 8 \implies b = 4 \]
2. Constant term:
\[ b + c = 3 \]
Substituting \( b = 4 \):
\[ 4 + c = 3 \implies c = 3 - 4 = -1 \]

### Final Values
Thus, we have:
\[ b = 4, \quad c = -1 \]

### Conclusion
The values of \( b \) and \( c \) are:
\[ b = 4, \quad c = -1 \]