`ab- mn + an - bm` = ?

Options

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

Answer

To factor the expression \( ab - mn + an - bm \), we can follow these steps:

1. **Rearrange the expression**:
We can rearrange the terms to group them in a way that makes factoring easier:
\[ ab + an - mn - bm \]

2. **Group the terms**:
Now, we can group the first two terms and the last two terms:
\[ (ab + an) + (-mn - bm) \]

3. **Factor out the common factors**:
- From the first group \( ab + an \), we can factor out \( a \):
\[ a(b + n) \]
- From the second group \( -mn - bm \), we can factor out \( -m \):
\[ -m(n + b) \]

Now, we rewrite the expression:
\[ a(b + n) - m(n + b) \]

4. **Notice that \( (n + b) \) can be rewritten**:
Since \( (n + b) \) is the same as \( (b + n) \), we can rewrite the expression as:
\[ a(b + n) - m(b + n) \]

5. **Factor out the common binomial factor**:
Now, we can factor out the common binomial \( (b + n) \):
\[ (b + n)(a - m) \]

Thus, the final factored form of the expression \( ab - mn + an - bm \) is:
\[ (b + n)(a - m) \]