Which of the following is not a quadratic equation?

Options

A. `2(x-1)^(2)=4x^(2)-2x+1`
B. `2x-x^(2)=x^(2)+5`
C. `(sqrt(2)x+sqrt(3))^(2)=3x^(2)-5x`
D. `(x^(2)+2x)^(2)=x^(4)+3+4x^(2)`

Explanation

To determine which of the following options is not a quadratic equation, we need to analyze each option based on the standard form of a quadratic equation, which is given by:

\[ ax^2 + bx + c = 0 \]

where \( a \neq 0 \) and \( a, b, c \) are constants.

### Step-by-Step Solution:

1. **Option 1: \( 2(x - 1)^2 = 4x^2 - 2x + 1 \)**

- Expand the left side using the identity \( (a - b)^2 = a^2 - 2ab + b^2 \):
\[ 2(x - 1)^2 = 2(x^2 - 2x + 1) = 2x^2 - 4x + 2 \]
- Set the equation:
\[ 2x^2 - 4x + 2 = 4x^2 - 2x + 1 \]
- Rearranging gives:
\[ 0 = 4x^2 - 2x + 1 - 2x^2 + 4x - 2 \]
\[ 0 = 2x^2 + 2x - 1 \]
- This is in the form \( ax^2 + bx + c = 0 \) where \( a = 2 \neq 0 \). Hence, this is a quadratic equation.

2. **Option 2: \( 2x - x^2 = x^2 + 5 \)**

- Rearranging gives:
\[ 2x - x^2 - x^2 - 5 = 0 \]
\[ -2x^2 + 2x - 5 = 0 \]
- This can be rewritten as:
\[ 2x^2 - 2x + 5 = 0 \]
- Here, \( a = 2 \neq 0 \), so this is also a quadratic equation.

3. **Option 3: \( \sqrt{2x} + \sqrt{3} = 3x^2 - 5x \)**

- Rearranging gives:
\[ 3x^2 - 5x - \sqrt{2x} - \sqrt{3} = 0 \]
- This is not in the standard form because of the term \( \sqrt{2x} \). Thus, this does not fit the quadratic equation format.

4. **Option 4: \( (x^2 + 2x)^2 = x^4 + 3 + 4x^2 \)**

- Expanding the left side:
\[ (x^2 + 2x)^2 = x^4 + 4x^3 + 4x^2 \]
- Setting the equation:
\[ x^4 + 4x^3 + 4x^2 = x^4 + 3 + 4x^2 \]
- Rearranging gives:
\[ 4x^3 + 4x^2 - 4x^2 - 3 = 0 \]
\[ 4x^3 - 3 = 0 \]
- This is a cubic equation, not a quadratic equation.

### Conclusion:
The option that is **not a quadratic equation** is **Option 3**: \( \sqrt{2x} + \sqrt{3} = 3x^2 - 5x \).