The product of two expression is `x^3 + x^2 - 44x - 84`. If the HCF of these two expressions is `x+6`, then their LCM will be:

Options

A. `(x+2)(x+7) `
B. `(x+2)(x-7) `
C. `(x-2)(x+7)`
D. `(x-2)(x-7)`

Answer

To find the LCM of two expressions given their product and HCF, we can use the relationship:

\[ \text{LCM} \times \text{HCF} = \text{Product of the two expressions} \]

Given:
- Product of the two expressions: \( P = x^3 + x^2 - 44x - 84 \)
- HCF: \( \text{HCF} = x + 6 \)

We need to find the LCM.

### Step 1: Set up the equation for LCM
Using the relationship mentioned above, we can express LCM as:

\[ \text{LCM} = \frac{P}{\text{HCF}} \]

### Step 2: Substitute the values
Substituting the values of \( P \) and HCF into the equation:

\[ \text{LCM} = \frac{x^3 + x^2 - 44x - 84}{x + 6} \]

### Step 3: Perform polynomial long division
Now, we will divide \( x^3 + x^2 - 44x - 84 \) by \( x + 6 \).

1. Divide the leading term \( x^3 \) by \( x \) to get \( x^2 \).
2. Multiply \( x^2 \) by \( x + 6 \) to get \( x^3 + 6x^2 \).
3. Subtract \( (x^3 + 6x^2) \) from \( (x^3 + x^2 - 44x - 84) \):

\[ (x^3 + x^2 - 44x - 84) - (x^3 + 6x^2) = -5x^2 - 44x - 84 \]

4. Bring down the next term: \( -5x^2 - 44x - 84 \).
5. Divide \( -5x^2 \) by \( x \) to get \( -5x \).
6. Multiply \( -5x \) by \( x + 6 \) to get \( -5x^2 - 30x \).
7. Subtract \( (-5x^2 - 30x) \) from \( (-5x^2 - 44x - 84) \):

\[ (-5x^2 - 44x - 84) - (-5x^2 - 30x) = -14x - 84 \]

8. Bring down the next term: \( -14x - 84 \).
9. Divide \( -14x \) by \( x \) to get \( -14 \).
10. Multiply \( -14 \) by \( x + 6 \) to get \( -14x - 84 \).
11. Subtract \( (-14x - 84) \) from \( (-14x - 84) \):

\[ (-14x - 84) - (-14x - 84) = 0 \]

### Step 4: Write the result
The result of the division is:

\[ \text{LCM} = x^2 - 5x - 14 \]

### Step 5: Factor the quadratic expression
Now, we can factor \( x^2 - 5x - 14 \):

1. Find two numbers that multiply to \(-14\) and add to \(-5\). These numbers are \(-7\) and \(2\).
2. Thus, we can write:

\[ x^2 - 5x - 14 = (x - 7)(x + 2) \]

### Final Answer
The LCM of the two expressions is:

\[ \text{LCM} = (x - 7)(x + 2) \]

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