If `x^(3) - 5x^(2) - 2x - 6 = m(x^(2) + 2x + 1) + (11x + 1)`. Then `m=?`

Options

A. `x + 7`
B. `x - 3`
C. `x - 7`
D. `x + 3`

Answer

To solve the equation \( x^3 - 5x^2 - 2x - 6 = m(x^2 + 2x + 1) + (11x + 1) \) for \( m \), we will follow these steps:

### Step 1: Rearrange the equation
We start by rearranging the equation to isolate the terms involving \( m \):
\[ x^3 - 5x^2 - 2x - 6 - (11x + 1) = m(x^2 + 2x + 1) \]
This simplifies to:
\[ x^3 - 5x^2 - 13x - 7 = m(x^2 + 2x + 1) \]

### Step 2: Divide the polynomial
Next, we need to divide the left-hand side polynomial \( x^3 - 5x^2 - 13x - 7 \) by the polynomial \( x^2 + 2x + 1 \).

### Step 3: Perform polynomial long division
1. Divide the leading term \( x^3 \) by \( x^2 \) to get \( x \).
2. Multiply \( x \) by \( x^2 + 2x + 1 \):
\[ x(x^2 + 2x + 1) = x^3 + 2x^2 + x \]
3. Subtract this from \( x^3 - 5x^2 - 13x - 7 \):
\[ (x^3 - 5x^2 - 13x - 7) - (x^3 + 2x^2 + x) = -7x^2 - 14x - 7 \]

4. Now, divide the leading term \( -7x^2 \) by \( x^2 \) to get \( -7 \).
5. Multiply \( -7 \) by \( x^2 + 2x + 1 \):
\[ -7(x^2 + 2x + 1) = -7x^2 - 14x - 7 \]
6. Subtract this from \( -7x^2 - 14x - 7 \):
\[ (-7x^2 - 14x - 7) - (-7x^2 - 14x - 7) = 0 \]

### Step 4: Conclusion
Since the remainder is 0, we have:
\[ x^3 - 5x^2 - 13x - 7 = (x - 7)(x^2 + 2x + 1) \]
Thus, we find that:
\[ m = x - 7 \]

### Final Answer
The value of \( m \) is:
\[ \boxed{x - 7} \]