The expression
`x ^(2) - y ^(2) + x + y - z ^(2) + 2 yz - z` has one factors, which is

Options

A. `x + y - z +1`
B. `x - y - z +1`
C. `y -x + z`
D. `x - y + z +1`

Answer

To factor the expression \( x^2 - y^2 + x + y - z^2 + 2yz - z \), we can follow these steps:

### Step 1: Rearrange the expression
Start by rearranging the expression to group similar terms together:
\[ x^2 - y^2 + x + y - z^2 + 2yz - z = x^2 + x - z^2 + 2yz - y^2 + y - z \]

### Step 2: Recognize the difference of squares
Notice that \( x^2 - y^2 \) can be factored using the difference of squares:
\[ x^2 - y^2 = (x - y)(x + y) \]
So we can rewrite the expression as:
\[ (x - y)(x + y) + x + y - z^2 + 2yz - z \]

### Step 3: Factor the quadratic in \( z \)
Next, we can focus on the terms involving \( z \):
\[ -z^2 + 2yz - z = - (z^2 - 2yz + z) = - (z^2 - 2yz + y^2 - y^2 + z) \]
This can be rearranged to:
\[ -((z - y)^2 + (z - y)) \]

### Step 4: Combine the factors
Now, we can combine everything together:
\[ = (x - y)(x + y) + (x + y - (z - y)^2) \]
This simplifies to:
\[ = (x + y - z)(x - y + z + 1) \]

### Step 5: Identify the factors
From the above steps, we can see that the expression can be factored as:
\[ (x + y - z)(x - y + z + 1) \]

Thus, one of the factors of the expression \( x^2 - y^2 + x + y - z^2 + 2yz - z \) is:
\[ \boxed{x + y - z} \]