Which of the following is a quadratic equation ?

Options

A. `x^(2) + 2x + 1 = (4 - x)^(2) +3 `
B. `- 2x^(2) = (5 - x) (2x- (2)/(5))`
C. `( k + 1) x^(2) + (3)/(2) x = 7` where k = -1
D. `x^(3) - x^(2) = (x -1)^(3)`

Answer

To determine which of the given options is a quadratic equation, we need to analyze each option based on the definition of a quadratic equation. A quadratic equation is generally in the form \( ax^2 + bx + c = 0 \), where \( a \), \( b \), and \( c \) are constants, and \( a \neq 0 \).

### Step-by-Step Solution:

1. **Understanding the Definition**:
- A quadratic equation is defined as an equation of the form \( ax^2 + bx + c = 0 \).
- Here, \( a \) is the coefficient of \( x^2 \), \( b \) is the coefficient of \( x \), and \( c \) is the constant term.

2. **Analyzing Option 1**:
- The equation is given as \( x^2 + 2x + 1 = 4 - x^2 + 3 \).
- Rearranging gives \( x^2 + 2x + 1 + x^2 - 4 - 3 = 0 \).
- This simplifies to \( 2x^2 + 2x - 6 = 0 \).
- This is a quadratic equation since it has \( x^2 \) as the highest power.

3. **Analyzing Option 2**:
- The equation is \( -2x^2 = 5 - \frac{x(2x - 2)}{5} \).
- After simplifying, we find that the \( x^2 \) terms cancel out.
- Thus, this does not represent a quadratic equation.

4. **Analyzing Option 3**:
- The equation is \( (k + 1)x^2 + \frac{3}{2}x = 7 \).
- Substituting \( k = -1 \) gives \( 0 \cdot x^2 + \frac{3}{2}x - 7 = 0 \).
- Since the \( x^2 \) term is absent, this is not a quadratic equation.

5. **Analyzing Option 4**:
- The equation is \( x^3 - x^2 = (x - 1)^3 \).
- Expanding \( (x - 1)^3 \) gives \( x^3 - 3x^2 + 3x - 1 \).
- Rearranging gives \( x^3 - x^2 - (x^3 - 3x^2 + 3x - 1) = 0 \).
- This simplifies to \( 2x^2 - 3x + 1 = 0 \), which is a quadratic equation.

### Conclusion:
- **Option 1** and **Option 4** are quadratic equations.
- **Option 2** and **Option 3** are not quadratic equations.

### Final Answer:
The quadratic equations from the options are **Option 1** and **Option 4**.