`(a + 1)(a - 1)(a^(2) + 1) = ?`

Options

A. `(a^(4) - 2a^(2) - 1)`
B. `(a^(4) - a^(2) - 1)`
C. `(a^(4) - 1)`
D. `(a^(4) + 1)`

Answer

To solve the expression \((a + 1)(a - 1)(a^{2} + 1)\), we can follow these steps:

### Step 1: Apply the Difference of Squares Formula
We recognize that \((a + 1)(a - 1)\) can be simplified using the difference of squares formula:
\[ (a + b)(a - b) = a^2 - b^2 \]
In this case, \(b = 1\). Therefore, we can write:
\[ (a + 1)(a - 1) = a^2 - 1^2 = a^2 - 1 \]

### Step 2: Substitute Back into the Expression
Now we substitute this result back into the original expression:
\[ (a + 1)(a - 1)(a^{2} + 1) = (a^2 - 1)(a^2 + 1) \]

### Step 3: Apply the Difference of Squares Again
Next, we can apply the difference of squares formula again to \((a^2 - 1)(a^2 + 1)\):
\[ (a^2 + b)(a^2 - b) = a^4 - b^2 \]
Here, \(b = 1\). Thus, we have:
\[ (a^2 - 1)(a^2 + 1) = a^4 - 1^2 = a^4 - 1 \]

### Final Result
Putting it all together, we find:
\[ (a + 1)(a - 1)(a^{2} + 1) = a^4 - 1 \]

### Summary
The final answer is:
\[ \boxed{a^4 - 1} \]

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