Factorize `a-b-a^2+b^2`

Answer

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

### Step 1: Rearrange the expression
Rearranging the terms can help us see the factors more clearly:
\[ a - a^2 + b^2 - b \]

### Step 2: Group the terms
Now, we can group the terms:
\[ (a - a^2) + (b^2 - b) \]

### Step 3: Factor out common terms
From the first group \( a - a^2 \), we can factor out \( a \):
\[ a(1 - a) \]
From the second group \( b^2 - b \), we can factor out \( b \):
\[ b(b - 1) \]
So we have:
\[ a(1 - a) + b(b - 1) \]

### Step 4: Recognize a pattern
Notice that \( b(b - 1) \) can be rewritten as \( -b(1 - b) \):
\[ a(1 - a) - b(1 - b) \]

### Step 5: Factor using the difference of squares
Now we can factor this expression as a difference of squares:
\[ (1 - a)(a + b) \]

### Step 6: Final expression
Thus, the factorized form of the expression \( a - b - a^2 + b^2 \) is:
\[ (1 - a)(a + b) \]