If `x ^(2) +bx+b` is a factor of `x ^(3) +2x ^(2)+2x + c (c ne 0),` then `b+c` is :

Options

A. 2
B. `-1`
C. 0
D. `-2`

Answer

To solve the problem, we need to determine the values of \( b \) and \( c \) such that \( x^2 + bx + b \) is a factor of \( x^3 + 2x^2 + 2x + c \).

### Step-by-step Solution:

1. **Set up the division**: Since \( x^2 + bx + b \) is a factor of \( x^3 + 2x^2 + 2x + c \), we can perform polynomial long division. We will divide \( x^3 + 2x^2 + 2x + c \) by \( x^2 + bx + b \).

2. **First term of the quotient**: Divide the leading term of the dividend \( x^3 \) by the leading term of the divisor \( x^2 \) to get \( x \). Multiply \( x \) by \( x^2 + bx + b \):
\[ x(x^2 + bx + b) = x^3 + bx^2 + bx \]

3. **Subtract**: Subtract this from the original polynomial:
\[ (x^3 + 2x^2 + 2x + c) - (x^3 + bx^2 + bx) = (2 - b)x^2 + (2 - b)x + c \]

4. **Second term of the quotient**: Now we need to divide \( (2 - b)x^2 \) by \( x^2 \) to get \( 2 - b \). Multiply \( 2 - b \) by \( x^2 + bx + b \):
\[ (2 - b)(x^2 + bx + b) = (2 - b)x^2 + (2 - b)bx + (2 - b)b \]

5. **Subtract again**: Subtract this from the current polynomial:
\[ ((2 - b)x^2 + (2 - b)x + c) - ((2 - b)x^2 + (2 - b)bx + (2 - b)b) = (2 - b - (2 - b)b)x + (c - (2 - b)b) \]
This simplifies to:
\[ (2 - b - (2 - b)b)x + (c - (2 - b)b) \]

6. **Set the remainder to zero**: For \( x^2 + bx + b \) to be a factor, the remainder must be zero. Thus:
- Coefficient of \( x \): \( 2 - b - (2 - b)b = 0 \)
- Constant term: \( c - (2 - b)b = 0 \)

7. **Solve for \( b \)**:
From \( 2 - b - (2 - b)b = 0 \):
\[ 2 - b - 2b + b^2 = 0 \implies b^2 - 3b + 2 = 0 \]
Factoring gives:
\[ (b - 1)(b - 2) = 0 \implies b = 1 \text{ or } b = 2 \]

8. **Find corresponding \( c \)**:
- If \( b = 1 \):
\[ c - (2 - 1) \cdot 1 = 0 \implies c - 1 = 0 \implies c = 1 \]
- If \( b = 2 \):
\[ c - (2 - 2) \cdot 2 = 0 \implies c - 0 = 0 \implies c = 0 \text{ (not valid since } c \neq 0\text{)} \]

9. **Final values**: Thus, the valid values are \( b = 1 \) and \( c = 1 \).

10. **Calculate \( b + c \)**:
\[ b + c = 1 + 1 = 2 \]

### Final Answer:
The value of \( b + c \) is \( 2 \).