Factorise :
`a b^(2) - (a-1) b-1`.

Answer

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

### Step 1: Rewrite the expression
Start with the expression:
\[ ab^2 - (a-1)b - 1 \]
Distributing the negative sign in the second term gives:
\[ ab^2 - ab + b - 1 \]

### Step 2: Group the terms
Now, we can group the terms in pairs:
\[ (ab^2 - ab) + (b - 1) \]

### Step 3: Factor out the common terms
From the first group \( ab^2 - ab \), we can factor out \( ab \):
\[ ab(b - 1) \]
From the second group \( b - 1 \), we can factor out \( 1 \):
\[ 1(b - 1) \]
Now we can rewrite the expression as:
\[ ab(b - 1) + 1(b - 1) \]

### Step 4: Factor out the common binomial
Now, we notice that \( (b - 1) \) is common in both terms:
\[ (b - 1)(ab + 1) \]

### Final Answer
Thus, the factorised form of the expression \( ab^2 - (a-1)b - 1 \) is:
\[ (b - 1)(ab + 1) \]

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