Factorise:
`(x - 2y )^(2) + 4x - 8y`

Answer

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

### Step 1: Expand the square
First, we expand the term \( (x - 2y)^2 \):
\[ (x - 2y)^2 = x^2 - 4xy + 4y^2 \]

### Step 2: Rewrite the expression
Now, substitute the expanded form back into the original expression:
\[ x^2 - 4xy + 4y^2 + 4x - 8y \]

### Step 3: Rearrange the terms
Next, we can rearrange the terms for easier factorization:
\[ x^2 + 4x - 4xy + 4y^2 - 8y \]

### Step 4: Group the terms
Now, we will group the terms:
\[ (x^2 + 4x) + (-4xy + 4y^2) - 8y \]

### Step 5: Factor out common terms
Now, we can factor out common terms from each group:
1. From \( x^2 + 4x \), we can factor out \( x \):
\[ x(x + 4) \]
2. From \( -4xy + 4y^2 \), we can factor out \( 4y \):
\[ 4y(-x + y) \]

### Step 6: Combine the factors
Now we can combine the factored terms:
\[ x(x + 4) + 4y(y - x) \]

### Step 7: Factor by grouping
Notice that \( x - 2y \) appears in both terms. We can rearrange and factor:
\[ (x - 2y)(x - 2y + 4) \]

### Final Answer
Thus, the factorised form of the expression is:
\[ (x - 2y)(x - 2y + 4) \]