Factorize the algebraic expression: x² - 1 - 2a - a²

To factor the expression \(x² - 1 - 2a - a²\), we can start by rearranging the terms for clarity.

Rearranging the Expression

The expression can be rewritten as:

• \(x² - (a² + 2a + 1)\)

Recognizing a Perfect Square

Notice that \(a² + 2a + 1\) is a perfect square trinomial. It can be factored as:

• \((a + 1)²\)

Final Factorization

Substituting this back into the expression gives:

• \(x² - (a + 1)²\)

This is now a difference of squares, which can be factored further:

• \((x - (a + 1))(x + (a + 1))\)

Result

The fully factored form of the original expression is:

(x - (a + 1))(x + (a + 1))