Which of the following is equivalent to `2b^(2) c^(2) + 2c^(2) a^(2) + 2a^(2) b^(2) - a^(4) - b^(4) - c^(4)`?

Options

A. `(a+ b + c) (a - b + c) (a+ b - c)(-a + b + c)`
B. `(a+ b + c) (a - b + c) (a+ b - c)`
C. `(a - b - c) (-a+ b + c) (a+ b - c)`
D. none of these

Answer

To solve the expression \(2b^2c^2 + 2c^2a^2 + 2a^2b^2 - a^4 - b^4 - c^4\), we can follow these steps:

### Step 1: Group the Terms
We start by grouping the terms in a way that allows us to factor them more easily. The expression can be rearranged as follows:

\[ (2b^2c^2 + 2c^2a^2 + 2a^2b^2) - (a^4 + b^4 + c^4) \]

### Step 2: Factor out the Common Terms
Notice that in the first part, we can factor out a 2:

\[ 2(b^2c^2 + c^2a^2 + a^2b^2) - (a^4 + b^4 + c^4) \]

### Step 3: Recognize the Sum of Squares
The expression \(a^4 + b^4 + c^4\) can be rewritten using the identity for the sum of squares. We know that:

\[ a^4 + b^4 + c^4 = (a^2 + b^2 + c^2)^2 - 2(a^2b^2 + b^2c^2 + c^2a^2) \]

### Step 4: Substitute the Identity
Substituting this identity into our expression gives:

\[ 2(b^2c^2 + c^2a^2 + a^2b^2) - \left((a^2 + b^2 + c^2)^2 - 2(b^2c^2 + c^2a^2 + a^2b^2)\right) \]

### Step 5: Simplify the Expression
Now, let's simplify this expression:

\[ 2(b^2c^2 + c^2a^2 + a^2b^2) - (a^2 + b^2 + c^2)^2 + 2(b^2c^2 + c^2a^2 + a^2b^2) \]

Combining like terms results in:

\[ 4(b^2c^2 + c^2a^2 + a^2b^2) - (a^2 + b^2 + c^2)^2 \]

### Step 6: Final Factorization
This can be factored further. The expression can be recognized as:

\[ -(a^2 + b^2 + c^2)^2 + 4(b^2c^2 + c^2a^2 + a^2b^2) \]

This can be rewritten as:

\[ -(a^2 - b^2 - c^2)^2 \]

### Conclusion
Thus, the final equivalent expression is:

\[ -(a^2 - b^2 - c^2)^2 \]