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

Options

A. `(1+ a-b) (1+ a+b)`
B. `(1 + a+ b) (1- a +b)`
C. `(1 + a+b) (1-a-b)`
D. `(1+ a- b) (1 - a+b)`

Answer

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

### Step 1: Rewrite the expression
Start with the original expression:
\[ 1 - 2ab - (a^2 + b^2) \]
We can rewrite it as:
\[ 1 - 2ab - a^2 - b^2 \]

### Step 2: Rearrange the terms
Now, we can rearrange the terms:
\[ 1 - a^2 - b^2 - 2ab \]

### Step 3: Recognize the pattern
Notice that the expression \(a^2 + b^2 + 2ab\) is the expansion of \((a + b)^2\). Therefore, we can rewrite our expression:
\[ 1 - (a^2 + b^2 + 2ab) = 1 - (a + b)^2 \]

### Step 4: Apply the difference of squares formula
Now, we can use the difference of squares formula, which states that \(x^2 - y^2 = (x + y)(x - y)\). Here, we can let \(x = 1\) and \(y = (a + b)\):
\[ 1 - (a + b)^2 = (1 + (a + b))(1 - (a + b)) \]

### Step 5: Write the final factorization
Thus, the final factorization of the expression is:
\[ (1 + a + b)(1 - a - b) \]

### Summary of the solution
The expression \(1 - 2ab - (a^2 + b^2)\) can be factored as:
\[ (1 + a + b)(1 - a - b) \]