`(4x^3y - 6x^2 y^2 + 4xy^3 - y^4)` can be expressed as :

Options

A. `(x - y)^4 - x^4`
B. `(x + y)^4 - y^4`
C. `(x + y)^4 - x^4`
D. `x^4 - (x - y)^4`

Answer

To express the polynomial \(4x^3y - 6x^2y^2 + 4xy^3 - y^4\) in a different form, we can follow these steps:

### Step 1: Identify the Expression
The expression we are working with is:
\[ 4x^3y - 6x^2y^2 + 4xy^3 - y^4 \]

### Step 2: Factor by Grouping
We can group the terms in pairs:
\[ (4x^3y - 6x^2y^2) + (4xy^3 - y^4) \]

### Step 3: Factor Out Common Terms
From the first group \(4x^3y - 6x^2y^2\), we can factor out \(2x^2y\):
\[ 2x^2y(2x - 3y) \]
From the second group \(4xy^3 - y^4\), we can factor out \(y^3\):
\[ y^3(4x - y) \]

### Step 4: Combine the Factored Terms
Now we rewrite the expression as:
\[ 2x^2y(2x - 3y) + y^3(4x - y) \]

### Step 5: Look for Further Factorization
At this point, we can check if we can factor further. However, we can also evaluate the expression for specific values of \(x\) and \(y\) to see if it matches any of the options given in the question.

### Step 6: Substitute Values
Let’s substitute \(x = 0\) and \(y = 1\) into the original expression:
\[ 4(0)^3(1) - 6(0)^2(1)^2 + 4(0)(1)^3 - (1)^4 = 0 - 0 + 0 - 1 = -1 \]

### Step 7: Substitute Values in Options
Now we substitute \(x = 0\) and \(y = 1\) into each of the options provided to find which one equals \(-1\).

1. **Option 1:** \(0 - 1^4 - 0 = 0 - 1 - 0 = -1\)
2. **Option 2:** \(0 + 1^4 - 1^4 = 0 + 1 - 1 = 0\)
3. **Option 3:** \(0 + 1^4 - 0 = 0 + 1 - 0 = 1\)
4. **Option 4:** \(0 - 0 - 1^4 = 0 - 0 - 1 = -1\)

### Conclusion
The options that yield \(-1\) are Option 1 and Option 4. Therefore, the expression can be expressed in the form of Option 4.