Fractorise:
`ab ^(2) + (a -1) b-1`

Answer

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

### Step 1: Rewrite the expression
The expression is already given as:
\[ ab^2 + (a - 1)b - 1 \]

### Step 2: Identify the coefficients
We can identify the coefficients of the quadratic expression in terms of \( b \):
- The coefficient of \( b^2 \) is \( a \).
- The coefficient of \( b \) is \( (a - 1) \).
- The constant term is \( -1 \).

### Step 3: Use the quadratic formula
We can factor the quadratic expression \( ab^2 + (a - 1)b - 1 \) using the quadratic formula, which is given by:
\[ b = \frac{-B \pm \sqrt{B^2 - 4AC}}{2A} \]
where \( A = a \), \( B = (a - 1) \), and \( C = -1 \).

### Step 4: Calculate the discriminant
First, we calculate the discriminant \( B^2 - 4AC \):
\[ B^2 = (a - 1)^2 = a^2 - 2a + 1 \]
\[ 4AC = 4 \cdot a \cdot (-1) = -4a \]
Thus, the discriminant is:
\[ D = (a^2 - 2a + 1) + 4a = a^2 + 2a + 1 = (a + 1)^2 \]

### Step 5: Find the roots
Now we can find the roots using the quadratic formula:
\[ b = \frac{-(a - 1) \pm (a + 1)}{2a} \]
Calculating the two cases:
1. \( b_1 = \frac{-(a - 1) + (a + 1)}{2a} = \frac{2}{2a} = \frac{1}{a} \)
2. \( b_2 = \frac{-(a - 1) - (a + 1)}{2a} = \frac{-2a}{2a} = -1 \)

### Step 6: Write the factorized form
Thus, the expression can be factorized as:
\[ ab^2 + (a - 1)b - 1 = a(b - \frac{1}{a})(b + 1) \]
or
\[ = a(b - 1)(b + 1) \]

### Final Factorized Expression
The final factorized form of the expression is:
\[ a(b - 1)(b + 1) \]

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