The HCF of `(x^(2)-4)(x^(2)-5x-6) and (x^(2)+x-6)` is :

Options

A. 1
B. `x+2`
C. `x-2`
D. `(x^(2)+x-6)`

Answer

To find the HCF (Highest Common Factor) of the polynomials \( (x^2 - 4)(x^2 - 5x - 6) \) and \( (x^2 + x - 6) \), we will follow these steps:

### Step 1: Factor the first polynomial \( (x^2 - 4)(x^2 - 5x - 6) \)

1. **Factor \( x^2 - 4 \)**:
- This is a difference of squares:
\[ x^2 - 4 = (x - 2)(x + 2) \]

2. **Factor \( x^2 - 5x - 6 \)**:
- We need to find two numbers that multiply to \(-6\) (the constant term) and add to \(-5\) (the coefficient of \(x\)). The numbers are \(-6\) and \(1\):
\[ x^2 - 5x - 6 = (x - 6)(x + 1) \]

3. **Combine the factors**:
- Thus, we can write:
\[ (x^2 - 4)(x^2 - 5x - 6) = (x - 2)(x + 2)(x - 6)(x + 1) \]

### Step 2: Factor the second polynomial \( x^2 + x - 6 \)

1. **Factor \( x^2 + x - 6 \)**:
- We need to find two numbers that multiply to \(-6\) and add to \(1\). The numbers are \(3\) and \(-2\):
\[ x^2 + x - 6 = (x - 2)(x + 3) \]

### Step 3: Identify common factors

Now we have:
- From the first polynomial: \( (x - 2)(x + 2)(x - 6)(x + 1) \)
- From the second polynomial: \( (x - 2)(x + 3) \)

### Step 4: Find the HCF

The common factor in both factorizations is \( (x - 2) \).

### Conclusion

Thus, the HCF of \( (x^2 - 4)(x^2 - 5x - 6) \) and \( (x^2 + x - 6) \) is:
\[ \text{HCF} = x - 2 \]

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