Factorise :
`x^(2) + (a^(2) -1)/(a)x-1`

Answer

To factorise the expression \( x^2 + \frac{a^2 - 1}{a} x - 1 \), we will follow these steps:

### Step 1: Rewrite the expression
Start with the given expression:
\[ x^2 + \frac{a^2 - 1}{a} x - 1 \]

### Step 2: Clear the fraction
To eliminate the fraction, we can multiply the entire expression by \( a \):
\[ a \left( x^2 + \frac{a^2 - 1}{a} x - 1 \right) = ax^2 + (a^2 - 1)x - a \]

### Step 3: Rearrange the expression
Now we have:
\[ ax^2 + (a^2 - 1)x - a \]

### Step 4: Factor by grouping
We will look for two numbers that multiply to \( a \cdot (-a) = -a^2 \) and add to \( a^2 - 1 \). These numbers are \( a \) and \( -1 \). Rewrite the middle term:
\[ ax^2 + ax - x - a \]

### Step 5: Group the terms
Group the first two terms and the last two terms:
\[ (a x^2 + ax) + (-x - a) \]

### Step 6: Factor out common terms
Factor out \( x \) from the first group and \( -1 \) from the second group:
\[ x(a x + a) - 1(x + a) \]

### Step 7: Factor out the common binomial
Now we can factor out the common binomial \( (x + a) \):
\[ (x + a)(ax - 1) \]

### Step 8: Final expression
Thus, the factorised form of the original expression is:
\[ (x + a)(ax - 1) \]