Let C and K be constants. If `x^2 + Kx + 5` factors into `(x + 1)(x + C)`, the value of K is

Options

A. `0`
B. `5`
C. `6`
D. `8`

Answer

To solve the problem, we need to find the value of \( K \) given that the quadratic expression \( x^2 + Kx + 5 \) factors into \( (x + 1)(x + C) \).

### Step-by-step Solution:

1. **Set up the equation**: We start with the equation given in the problem:
\[ x^2 + Kx + 5 = (x + 1)(x + C) \]

2. **Expand the right-hand side**: We will expand the product on the right side:
\[ (x + 1)(x + C) = x^2 + Cx + x + 1 = x^2 + (C + 1)x + 1 \]

3. **Equate coefficients**: Now we can equate the coefficients from both sides of the equation:
\[ x^2 + Kx + 5 = x^2 + (C + 1)x + 1 \]

From this, we can derive two equations by comparing coefficients:
- Coefficient of \( x \): \( K = C + 1 \) (Equation 1)
- Constant term: \( 5 = 1 \) (This is incorrect, we should have \( 5 = C \)) (Equation 2)

4. **Solve for \( C \)**: From Equation 2, we have:
\[ C = 5 \]

5. **Substitute \( C \) into Equation 1**: Now we substitute \( C \) back into Equation 1 to find \( K \):
\[ K = C + 1 = 5 + 1 = 6 \]

6. **Final answer**: Therefore, the value of \( K \) is:
\[ \boxed{6} \]