Factorise :
`a^(3) - 27b^(3) + 2a^(2)b - 6ab^(2)`

Answer

To factorise the expression \( a^3 - 27b^3 + 2a^2b - 6ab^2 \), we can follow these steps:

### Step 1: Rewrite the expression
We start with the expression:
\[ a^3 - 27b^3 + 2a^2b - 6ab^2 \]
Notice that \( 27b^3 \) can be rewritten as \( (3b)^3 \). Thus, we can express the equation as:
\[ a^3 - (3b)^3 + 2a^2b - 6ab^2 \]

### Step 2: Group the terms
Now, we can group the terms:
\[ a^3 - (3b)^3 + 2ab(a - 3b) \]
Here, we factor \( 2ab \) from the last two terms \( 2a^2b - 6ab^2 \).

### Step 3: Recognize the difference of cubes
We recognize that \( a^3 - (3b)^3 \) is a difference of cubes. The formula for the difference of cubes is:
\[ x^3 - y^3 = (x - y)(x^2 + xy + y^2) \]
Applying this to our expression, we have:
\[ x = a \quad \text{and} \quad y = 3b \]
Thus, we can write:
\[ a^3 - (3b)^3 = (a - 3b)(a^2 + a(3b) + (3b)^2) = (a - 3b)(a^2 + 3ab + 9b^2) \]

### Step 4: Combine the factored terms
Now, substituting back into our expression:
\[ (a - 3b)(a^2 + 3ab + 9b^2) + 2ab(a - 3b) \]
We can factor out \( (a - 3b) \):
\[ (a - 3b)(a^2 + 3ab + 9b^2 + 2ab) \]

### Step 5: Simplify the expression inside the parentheses
Now, we simplify the expression inside the parentheses:
\[ a^2 + 3ab + 2ab + 9b^2 = a^2 + 5ab + 9b^2 \]

### Final Factorization
Thus, the final factorization of the expression is:
\[ \boxed{(a - 3b)(a^2 + 5ab + 9b^2)} \]

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