Factorise :
` ( a^(2) + 1) b^(2) - b^(4) - a^(2)`

Answer

To factorise the expression \( (a^2 + 1)b^2 - b^4 - a^2 \), we will follow these steps:

### Step 1: Expand the expression
Start by distributing \( b^2 \) into the first part of the expression:
\[ (a^2 + 1)b^2 - b^4 - a^2 = a^2b^2 + b^2 - b^4 - a^2 \]

### Step 2: Rearrange the terms
Now, rearranging the terms gives:
\[ a^2b^2 - a^2 + b^2 - b^4 \]

### Step 3: Group the terms
Next, group the terms involving \( a^2 \) and the remaining terms:
\[ a^2(b^2 - 1) + b^2 - b^4 \]

### Step 4: Factor out common terms
Notice that \( b^2 - b^4 \) can be factored:
\[ b^2 - b^4 = b^2(1 - b^2) \]
So, the expression now looks like:
\[ a^2(b^2 - 1) + b^2(1 - b^2) \]

### Step 5: Factor out \( (b^2 - 1) \)
Now, we can factor out \( (b^2 - 1) \):
\[ (b^2 - 1)(a^2 - b^2) \]

### Step 6: Recognize the difference of squares
The term \( a^2 - b^2 \) can be factored further using the difference of squares:
\[ a^2 - b^2 = (a - b)(a + b) \]

### Step 7: Final factorization
Thus, the complete factorization of the original expression is:
\[ (b - 1)(b + 1)(a - b)(a + b) \]

### Final Answer
The factors of the expression \( (a^2 + 1)b^2 - b^4 - a^2 \) are:
\[ (b - 1)(b + 1)(a - b)(a + b) \]

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