The expression $x^2 + 2xy + y^2 - a^2 + 2ab - b^2$ has one factor as:

Q: The expression $x^2 + 2xy + y^2 - a^2 + 2ab - b^2$ has one factor as:

To determine why Option 2, $ x + y - a + b $, is a factor of the expression $ x^2 + 2xy + y^2 - a^2 + 2ab - b^2 $, we need to analyze the expression and factor it appropriately. ### Explanation for Option 2: The given expression is: \[ x^2 + 2xy + y^2 - a^2 + 2ab - b^2 \] This expression can be rewritten by grouping terms: \[ (x^2 + 2xy + y^2) - (a^2 - 2ab + b^2) \] Notice that both groups are perfect square trinomials: 1. $ x^2 + 2xy + y^2 $ is the expansion of $ (x + y)^2 $. 2. $ a^2 - 2ab + b^2 $ is the expansion of $ (a - b)^2 $. Thus, the expression becomes: \[ (x + y)^2 - (a - b)^2 \] This is a difference of squares, which can be factored using the identity $ A^2 - B^2 = (A - B)(A + B) $. Here, $ A = x + y $ and $ B = a - b $. Applying the difference of squares formula, we get: \[ (x + y - (a - b))(x + y + (a - b)) \] Simplifying the factors, we have: 1. $ x + y - a + b $ 2. $ x + y + a - b $ Thus, one of the factors is indeed $ x + y - a + b $, which corresponds to Option 2. ### Explanation for Incorrect Options: - **Option 1: $ x - y - a - b $** This option does not match any factor derived from the difference of squares. The expression $ x - y - a - b $ does not align with the factors obtained from the correct factorization process. - **Option 3: $ x - y - a + b $** Similar to Option 1, this expression does not correspond to any factor derived from the difference of squares. The signs and terms do not match the factors obtained from the correct factorization. - **Option 4: $ x + y + a + b $** While this option shares some terms with the correct factor, it does not match the factorization derived from the difference of squares. The presence of $ +a + b $ instead of the correct combination of terms makes it incorrect. In conclusion, Option 2, $ x + y - a + b $, is the correct factor of the given expression based on the factorization process using the difference of squares.

Question

Accepted Answer

To determine why Option 2, $ x + y - a + b $, is a factor of the expression $ x^2 + 2xy + y^2 - a^2 + 2ab - b^2 $, we need to analyze the expression and factor it appropriately.

### Explanation for Option 2:

The given expression is:

$$ x^2 + 2xy + y^2 - a^2 + 2ab - b^2 $$

This expression can be rewritten by grouping terms:

$$ (x^2 + 2xy + y^2) - (a^2 - 2ab + b^2) $$

Notice that both groups are perfect square trinomials:

1. $ x^2 + 2xy + y^2 $ is the expansion of $ (x + y)^2 $.
2. $ a^2 - 2ab + b^2 $ is the expansion of $ (a - b)^2 $.

Thus, the expression becomes:

$$ (x + y)^2 - (a - b)^2 $$

This is a difference of squares, which can be factored using the identity $ A^2 - B^2 = (A - B)(A + B) $. Here, $ A = x + y $ and $ B = a - b $.

Applying the difference of squares formula, we get:

$$ (x + y - (a - b))(x + y + (a - b)) $$

Simplifying the factors, we have:

1. $ x + y - a + b $
2. $ x + y + a - b $

Thus, one of the factors is indeed $ x + y - a + b $, which corresponds to Option 2.

### Explanation for Incorrect Options:

- **Option 1: $ x - y - a - b $**

This option does not match any factor derived from the difference of squares. The expression $ x - y - a - b $ does not align with the factors obtained from the correct factorization process.

- **Option 3: $ x - y - a + b $**

Similar to Option 1, this expression does not correspond to any factor derived from the difference of squares. The signs and terms do not match the factors obtained from the correct factorization.

- **Option 4: $ x + y + a + b $**

While this option shares some terms with the correct factor, it does not match the factorization derived from the difference of squares. The presence of $ +a + b $ instead of the correct combination of terms makes it incorrect.

In conclusion, Option 2, $ x + y - a + b $, is the correct factor of the given expression based on the factorization process using the difference of squares.