The factors of `1- (a ^(2) + b ^(2)) + a ^(2) b ^(2)` are :

Options

A. `(1 +a ) (1-a ) (1 + b ^(2))`
B. `(1 + a ^(2)) (1 + b ^(2))`
C. `(1 + a ^(2)) (1- b) (1 + b)`
D. `(1 - a) (1 + a) (1- b) (1 + b)`

Answer

To factor the expression \(1 - (a^2 + b^2) + a^2b^2\), we will follow these steps:

### Step 1: Rewrite the expression
Start with the original expression:
\[ 1 - (a^2 + b^2) + a^2b^2 \]
This can be rewritten as:
\[ 1 - a^2 - b^2 + a^2b^2 \]

### Step 2: Rearrange the terms
Rearranging the terms gives us:
\[ 1 - a^2 + a^2b^2 - b^2 \]

### Step 3: Group the terms
Now, we can group the terms:
\[ (1 - a^2) + (a^2b^2 - b^2) \]

### Step 4: Factor out common terms
From the first group, we can factor out \(1 - a^2\):
\[ 1 - a^2 = (1 - a)(1 + a) \]
From the second group, we can factor out \(b^2\):
\[ a^2b^2 - b^2 = b^2(a^2 - 1) = b^2(1 - a)(1 + a) \]

### Step 5: Combine the factors
Now, we can combine the factors:
\[ (1 - a)(1 + a) + b^2(1 - a)(1 + a) \]
Factoring out \((1 - a)(1 + a)\) gives:
\[ (1 - a)(1 + a)(1 + b^2) \]

### Final Factored Form
Thus, the final factored form of the expression \(1 - (a^2 + b^2) + a^2b^2\) is:
\[ (1 - a)(1 + a)(1 + b^2) \]