If the HCF of `x^(2)-x-6` and `x^(2)+9x+14` is `(x+m)`. Then the value of m is

Options

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

Answer

To find the value of \( m \) in the given problem, we need to find the HCF of the two polynomials \( x^2 - x - 6 \) and \( x^2 + 9x + 14 \). Let's go through the steps to solve this:

### Step 1: Factor the first polynomial \( x^2 - x - 6 \)

To factor \( x^2 - x - 6 \), we look for two numbers that multiply to \(-6\) (the constant term) and add to \(-1\) (the coefficient of \( x \)).

The numbers that satisfy this are \(-3\) and \(2\). Thus, we can factor the polynomial as follows:

\[ x^2 - x - 6 = (x - 3)(x + 2) \]

### Step 2: Factor the second polynomial \( x^2 + 9x + 14 \)

Next, we factor \( x^2 + 9x + 14 \). We need two numbers that multiply to \(14\) and add to \(9\).

The numbers that satisfy this are \(7\) and \(2\). Therefore, we can factor this polynomial as:

\[ x^2 + 9x + 14 = (x + 7)(x + 2) \]

### Step 3: Find the HCF of the two polynomials

Now that we have both polynomials factored, we can find the HCF. The factored forms are:

1. \( (x - 3)(x + 2) \)
2. \( (x + 7)(x + 2) \)

The common factor in both polynomials is \( (x + 2) \). Therefore, the HCF is:

\[ \text{HCF} = (x + 2) \]

### Step 4: Relate HCF to \( x + m \)

According to the problem, the HCF is given as \( (x + m) \). Since we found that the HCF is \( (x + 2) \), we can equate:

\[ x + m = x + 2 \]

### Step 5: Solve for \( m \)

From the equation \( x + m = x + 2 \), we can see that:

\[ m = 2 \]

### Final Answer

Thus, the value of \( m \) is:

\[ \boxed{2} \]