Factorise :
` (2a-3)^(2) - 2(2a-3) (a-1) +(a-1)^(2)`

Answer

To factorize the expression \( (2a-3)^{2} - 2(2a-3)(a-1) + (a-1)^{2} \), we can follow these steps:

### Step 1: Identify the structure of the expression
The expression resembles the form of a perfect square trinomial, which is given by:
\[ x^2 - 2xy + y^2 = (x - y)^2 \]
In our case, we can let:
- \( x = (2a - 3) \)
- \( y = (a - 1) \)

### Step 2: Rewrite the expression using \( x \) and \( y \)
Substituting \( x \) and \( y \) into the expression, we have:
\[ (2a - 3)^{2} - 2(2a - 3)(a - 1) + (a - 1)^{2} \]
This can be rewritten as:
\[ x^{2} - 2xy + y^{2} \]

### Step 3: Factor the expression
Using the identity for a perfect square trinomial, we can factor the expression as:
\[ (x - y)^{2} \]
Substituting back \( x \) and \( y \):
\[ (2a - 3 - (a - 1))^{2} \]

### Step 4: Simplify the expression inside the parentheses
Now, simplify \( 2a - 3 - (a - 1) \):
\[ 2a - 3 - a + 1 = 2a - a - 3 + 1 = a - 2 \]

### Step 5: Write the final factored form
Thus, the entire expression can be factored as:
\[ (a - 2)^{2} \]

### Final Answer
The factorized form of the expression is:
\[ (a - 2)^{2} \]