Factorise : `(m + a ) (n -b) - (m + b) (n -a )`

Options

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

Answer

To factorize the expression \((m + a)(n - b) - (m + b)(n - a)\), we will follow these steps:

### Step 1: Expand the expression
First, we will expand both products in the expression.

\[ (m + a)(n - b) = mn - mb + an - ab \]
\[ (m + b)(n - a) = mn - ma + bn - ab \]

Now, substituting these expansions back into the original expression:

\[ (m + a)(n - b) - (m + b)(n - a) = (mn - mb + an - ab) - (mn - ma + bn - ab) \]

### Step 2: Simplify the expression
Now, we will simplify the expression by distributing the negative sign:

\[ = mn - mb + an - ab - mn + ma - bn + ab \]

Notice that \(mn\) and \(-mn\) cancel each other out, and \(-ab\) and \(+ab\) also cancel each other out:

\[ = -mb + an + ma - bn \]

### Step 3: Rearrange the terms
Now we can rearrange the terms to group similar ones:

\[ = an + ma - mb - bn \]

### Step 4: Factor by grouping
Now we will group the terms:

\[ = (an + ma) + (-mb - bn) \]

Now, we can factor out common factors from each group:

\[ = a(n + m) - b(m + n) \]

### Step 5: Factor out the common binomial
Notice that \(n + m\) and \(m + n\) are the same, so we can factor that out:

\[ = (a - b)(m + n) \]

### Final Result
Thus, the factorized form of the expression \((m + a)(n - b) - (m + b)(n - a)\) is:

\[ \boxed{(a - b)(m + n)} \]

---