Among below expressions which one can be expressed as the difference of two cubic expressions?

Options

A. `x ^(3) + y ^(3) + 3x -2`
B. `x ^(2) + 2x ^(2) + 3x`
C. `x ^(3) + 3x ^(2) + 3x -2`
D. `x ^(3) + 3x ^(2) + 3x - 7`

Answer

To determine which expression can be expressed as the difference of two cubic expressions, we need to analyze the given options step by step.

### Step-by-Step Solution:

1. **Identify the Expressions**: We have the following expressions to analyze:
- Expression 1: \( x^3 + y^3 + 3x - 2 \)
- Expression 2: \( x^2 + 2x^2 + 3x \) (which simplifies to \( 3x^2 + 3x \))
- Expression 3: \( x^3 + 3x^2 + 3x - 2 \)
- Expression 4: \( x^3 + 3x^2 + 3x - 7 \)

2. **Check for Cubic Terms**: We need to check if the expressions can be represented as the difference of two cubic expressions. For that, we need to ensure that both expressions involved are cubic.

3. **Analyze Each Expression**:
- **Expression 1**: \( x^3 + y^3 + 3x - 2 \)
- This can be rearranged as \( x^3 - (-y^3 + 3x - 2) \). Here, \( x^3 \) and \( -y^3 \) are cubic terms.
- **Expression 2**: \( 3x^2 + 3x \)
- This expression does not contain a cubic term, hence it cannot be expressed as a difference of two cubic expressions.
- **Expression 3**: \( x^3 + 3x^2 + 3x - 2 \)
- This can be rearranged as \( x^3 - (-3x^2 - 3x + 2) \). Here, \( x^3 \) is cubic, but the second part \( -(-3x^2 - 3x + 2) \) is not cubic.
- **Expression 4**: \( x^3 + 3x^2 + 3x - 7 \)
- Similar to Expression 3, this can be rearranged as \( x^3 - (-3x^2 - 3x + 7) \). Again, \( x^3 \) is cubic, but the second part is not cubic.

4. **Conclusion**:
- The only expression that can be expressed as the difference of two cubic expressions is **Expression 1**: \( x^3 + y^3 + 3x - 2 \).

### Final Answer:
The expression that can be expressed as the difference of two cubic expressions is:
**\( x^3 + y^3 + 3x - 2 \)**.