Factorise : `(2x-3)^2-8x+12`

Answer

To factorise the expression `(2x - 3)^2 - 8x + 12`, we will follow these steps:

### Step 1: Expand the squared term
First, we expand the term `(2x - 3)^2`.

\[ (2x - 3)^2 = 4x^2 - 12x + 9 \]

### Step 2: Substitute back into the expression
Now, we substitute this back into the original expression:

\[ 4x^2 - 12x + 9 - 8x + 12 \]

### Step 3: Combine like terms
Next, we combine the like terms:

\[ 4x^2 - 12x - 8x + 9 + 12 = 4x^2 - 20x + 21 \]

### Step 4: Factor the quadratic expression
Now we need to factor the quadratic expression \(4x^2 - 20x + 21\). We look for two numbers that multiply to \(4 \times 21 = 84\) and add to \(-20\). The numbers \(-14\) and \(-6\) fit this requirement.

We can rewrite the expression as:

\[ 4x^2 - 14x - 6x + 21 \]

### Step 5: Group the terms
Now we group the terms:

\[ (4x^2 - 14x) + (-6x + 21) \]

### Step 6: Factor out the common factors
From the first group \(4x^2 - 14x\), we can factor out \(2x\):

\[ 2x(2x - 7) \]

From the second group \(-6x + 21\), we can factor out \(-3\):

\[ -3(2x - 7) \]

### Step 7: Combine the factors
Now we can combine the two groups:

\[ 2x(2x - 7) - 3(2x - 7) = (2x - 3)(2x - 7) \]

### Final Result
Thus, the factorised form of the expression is:

\[ (2x - 3)(2x - 7) \]

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