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

Answer

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

### Step 1: Eliminate the fraction by multiplying through by \( a \)

We start with the expression:
\[ x^2 - x\left(\frac{a^2 - 1}{a}\right) - 1 \]

To eliminate the fraction, we multiply the entire expression by \( a \):
\[ a \cdot (x^2) - a \cdot \left(x\left(\frac{a^2 - 1}{a}\right)\right) - a \cdot 1 \]

This simplifies to:
\[ ax^2 - x(a^2 - 1) - a \]

### Step 2: Rewrite the expression

Now we can rewrite the expression as:
\[ ax^2 - x(a^2 - 1) - a \]

### Step 3: Rearrange the expression

We can rearrange it to group the terms:
\[ ax^2 - x(a^2 - 1) - a = ax^2 - (a^2 - 1)x - a \]

### Step 4: Factor by grouping

Next, we will factor by grouping. We can look for two numbers that multiply to \( -a \cdot (a^2 - 1) \) and add to \( 0 \).

The expression can be factored as:
\[ a(x^2 - x \cdot \frac{(a^2 - 1)}{a} - 1) \]

### Step 5: Factor the quadratic expression

Now we can factor the quadratic expression \( x^2 - x\left(\frac{(a^2 - 1)}{a}\right) - 1 \).

This can be factored as:
\[ (x - 1)(x + \frac{(a^2 - 1)}{a}) \]

### Final Answer

Thus, the fully factored form of the original expression is:
\[ a(x - 1)(x + \frac{(a^2 - 1)}{a}) \]