Factorise:
`4n^(2) - 8n +3`

Answer

To factorise the expression \(4n^2 - 8n + 3\), we can follow these steps:

### Step 1: Identify the coefficients
The given expression is \(4n^2 - 8n + 3\). Here, the coefficients are:
- Coefficient of \(n^2\) (let's call it \(a\)) = 4
- Coefficient of \(n\) (let's call it \(b\)) = -8
- Constant term (let's call it \(c\)) = 3

### Step 2: Multiply \(a\) and \(c\)
Next, we multiply \(a\) and \(c\):
\[ a \cdot c = 4 \cdot 3 = 12 \]

### Step 3: Find two numbers that multiply to \(12\) and add to \(-8\)
We need to find two numbers that multiply to \(12\) and add to \(-8\). The numbers that satisfy this condition are \(-6\) and \(-2\) because:
\[ -6 \cdot -2 = 12 \quad \text{and} \quad -6 + -2 = -8 \]

### Step 4: Rewrite the middle term
Now, we rewrite the expression by splitting the middle term \(-8n\) using the two numbers found:
\[ 4n^2 - 6n - 2n + 3 \]

### Step 5: Group the terms
Next, we group the terms:
\[ (4n^2 - 6n) + (-2n + 3) \]

### Step 6: Factor out the common terms in each group
Now, we factor out the common factors from each group:
- From the first group \(4n^2 - 6n\), we can factor out \(2n\):
\[ 2n(2n - 3) \]
- From the second group \(-2n + 3\), we can factor out \(-1\):
\[ -1(2n - 3) \]

### Step 7: Combine the factors
Now, we can combine the factored terms:
\[ 2n(2n - 3) - 1(2n - 3) \]
This gives us:
\[ (2n - 1)(2n - 3) \]

### Final Answer
Thus, the factorised form of \(4n^2 - 8n + 3\) is:
\[ (2n - 1)(2n - 3) \]

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