Factorise : (i) `(a+b)^2 -11 (a+b) - 42`

Answer

To factorise the expression \((a+b)^2 - 11(a+b) - 42\), we can follow these steps:

### Step 1: Substitute \(a + b\) with \(x\)
Let \(x = a + b\). Then the expression becomes:
\[ x^2 - 11x - 42 \]

### Step 2: Factor the quadratic expression
We need to factor the quadratic expression \(x^2 - 11x - 42\). We are looking for two numbers that multiply to \(-42\) (the constant term) and add to \(-11\) (the coefficient of \(x\)).

### Step 3: Find the two numbers
The factors of \(-42\) that add up to \(-11\) are \(-14\) and \(3\):
- \(-14 \times 3 = -42\)
- \(-14 + 3 = -11\)

### Step 4: Rewrite the quadratic expression
We can rewrite the quadratic expression using the two numbers found:
\[ x^2 - 14x + 3x - 42 \]

### Step 5: Group the terms
Now, we group the terms:
\[ (x^2 - 14x) + (3x - 42) \]

### Step 6: Factor by grouping
Now, we can factor out the common factors from each group:
\[ x(x - 14) + 3(x - 14) \]

### Step 7: Factor out the common binomial
Now, we can factor out the common binomial \((x - 14)\):
\[ (x - 14)(x + 3) \]

### Step 8: Substitute back \(x\) with \(a + b\)
Finally, we substitute back \(x\) with \(a + b\):
\[ (a + b - 14)(a + b + 3) \]

### Final Answer
Thus, the factorised form of \((a+b)^2 - 11(a+b) - 42\) is:
\[ (a + b - 14)(a + b + 3) \]

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