Factorise by grouping method :
`a^(2) + (1)/(a^(2)) - 2 - 3a + (3)/(a)`

Answer

To factorise the expression \( a^2 + \frac{1}{a^2} - 2 - 3a + \frac{3}{a} \) by grouping, we will follow these steps:

### Step 1: Rewrite the expression
Start with the original expression:
\[ a^2 + \frac{1}{a^2} - 2 - 3a + \frac{3}{a} \]

### Step 2: Group the terms
Rearrange the expression to group similar terms together:
\[ (a^2 - 3a - 2) + \left(\frac{1}{a^2} + \frac{3}{a}\right) \]

### Step 3: Simplify the first group
The first group is \( a^2 - 3a - 2 \). We can factor this expression:
1. Find two numbers that multiply to \(-2\) (the constant term) and add to \(-3\) (the coefficient of \(a\)). The numbers are \(-4\) and \(1\).
2. Thus, we can write:
\[ a^2 - 3a - 2 = (a - 4)(a + 1) \]

### Step 4: Simplify the second group
Now, simplify the second group \( \frac{1}{a^2} + \frac{3}{a} \):
1. Rewrite it as:
\[ \frac{1 + 3a}{a^2} \]

### Step 5: Combine the groups
Now we can combine the two groups:
\[ (a - 4)(a + 1) + \frac{1 + 3a}{a^2} \]

### Step 6: Factor out the common term
Notice that both groups share a common factor of \( (a - 1/a) \):
1. Rewrite \( a - 1/a \) as \( a - \frac{1}{a} \).
2. The expression can be factored as:
\[ (a - \frac{1}{a})(a - 3) \]

### Final Factorization
Thus, the factorization of the given expression is:
\[ \left(a - \frac{1}{a}\right)\left(a - 3\right) \]

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