Factorise :
`(x-y)^(3) - 8x^(3)`

Answer

To factorise the expression \((x - y)^3 - 8x^3\), we can follow these steps:

### Step 1: Recognize the difference of cubes
The expression is in the form of \(a^3 - b^3\), where:
- \(a = (x - y)\)
- \(b = (2x)\)

### Step 2: Apply the difference of cubes formula
The difference of cubes can be factored using the formula:
\[ a^3 - b^3 = (a - b)(a^2 + ab + b^2) \]
Substituting \(a\) and \(b\):
\[ (x - y)^3 - (2x)^3 = ((x - y) - 2x)((x - y)^2 + (x - y)(2x) + (2x)^2) \]

### Step 3: Simplify \(a - b\)
Calculate \(a - b\):
\[ (x - y) - 2x = -y - x = -(x + y) \]

### Step 4: Expand \(a^2 + ab + b^2\)
Now, we need to calculate \(a^2 + ab + b^2\):
1. Calculate \(a^2\):
\[ (x - y)^2 = x^2 - 2xy + y^2 \]
2. Calculate \(ab\):
\[ (x - y)(2x) = 2x(x - y) = 2x^2 - 2xy \]
3. Calculate \(b^2\):
\[ (2x)^2 = 4x^2 \]

Now combine these results:
\[ a^2 + ab + b^2 = (x^2 - 2xy + y^2) + (2x^2 - 2xy) + 4x^2 \]
Combine like terms:
\[ = x^2 + 2x^2 + 4x^2 - 2xy - 2xy + y^2 = 7x^2 - 4xy + y^2 \]

### Step 5: Write the final factorization
Putting it all together, we have:
\[ (x - y)^3 - 8x^3 = -(x + y)(7x^2 - 4xy + y^2) \]

### Final Answer:
The factorised form of \((x - y)^3 - 8x^3\) is:
\[ -(x + y)(7x^2 - 4xy + y^2) \]