The HCF of two expressions is `3x ^(2) + 4x - 4 and ` their LCM is `3x ^(4) + 4x ^(3) - 7x ^(2) - 4x + 4.` One of the expressions is

Options

A. `(x +1) ( 3x ^(2) + 4x +4)`
B. `(x -1) ( 3x ^(2) + 4x - 4)`
C. `(x -1) (3x ^(2) + 4x + 4)`
D. `(x +1) (3x ^(2) + x - 4)`

Answer

To solve the problem, we need to find one of the expressions given the HCF and LCM of two expressions. Let's denote the two expressions as \( P(x) \) and \( Q(x) \).

### Step-by-step Solution:

1. **Identify the HCF and LCM**:
- Given HCF: \( H(x) = 3x^2 + 4x - 4 \)
- Given LCM: \( L(x) = 3x^4 + 4x^3 - 7x^2 - 4x + 4 \)

2. **Use the relationship between HCF and LCM**:
- The relationship between the two expressions, their HCF, and their LCM is given by:
\[ P(x) \cdot Q(x) = H(x) \cdot L(x) \]
- This means that the product of the two expressions is equal to the product of their HCF and LCM.

3. **Factor the LCM by the HCF**:
- Since \( H(x) \) is a factor of \( L(x) \), we can divide \( L(x) \) by \( H(x) \) to find the other factor.
- Perform polynomial long division of \( L(x) \) by \( H(x) \).

4. **Perform the polynomial long division**:
- Divide \( 3x^4 + 4x^3 - 7x^2 - 4x + 4 \) by \( 3x^2 + 4x - 4 \):
- The first term of the quotient is \( x^2 \) (since \( 3x^4 \div 3x^2 = x^2 \)).
- Multiply \( H(x) \) by \( x^2 \) and subtract from \( L(x) \).
- Continue this process until you reach a remainder.

5. **Find the quotient**:
- After performing the division, you will find that:
\[ L(x) = H(x) \cdot (x^2 - 1) \]
- The expression \( x^2 - 1 \) can be factored further into \( (x - 1)(x + 1) \).

6. **Form the expressions**:
- Now, we can express \( P(x) \) and \( Q(x) \) using \( H(x) \):
- One possible expression could be:
\[ P(x) = H(x) \cdot (x - 1) = (3x^2 + 4x - 4)(x - 1) \]
- The other expression could be:
\[ Q(x) = H(x) \cdot (x + 1) = (3x^2 + 4x - 4)(x + 1) \]

7. **Conclusion**:
- Therefore, one of the expressions is:
\[ P(x) = (3x^2 + 4x - 4)(x - 1) \]
- You can expand this if needed, but the expression \( 3x^2 + 4x - 4 \) is already given as one of the expressions.