Factorise :
`a^(3) + 3a^(2)b + 3ab^(2) + 2b^(3)`.

Answer

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

### Step 1: Rewrite the expression
We start with the expression:
\[ a^3 + 3a^2b + 3ab^2 + 2b^3 \]
We can split \( 2b^3 \) into \( b^3 + b^3 \):
\[ a^3 + 3a^2b + 3ab^2 + b^3 + b^3 \]

### Step 2: Group the terms
Now, we can group the first five terms:
\[ (a^3 + 3a^2b + 3ab^2 + b^3) + b^3 \]

### Step 3: Recognize the identity
Notice that the first four terms resemble the expansion of \( (a + b)^3 \):
\[ a^3 + 3a^2b + 3ab^2 + b^3 = (a + b)^3 \]
Thus, we can rewrite the expression as:
\[ (a + b)^3 + b^3 \]

### Step 4: Apply the sum of cubes formula
Now, we can use the identity for the sum of cubes, which states:
\[ x^3 + y^3 = (x + y)(x^2 - xy + y^2) \]
In our case, let \( x = (a + b) \) and \( y = b \):
\[ (a + b)^3 + b^3 = ((a + b) + b)((a + b)^2 - (a + b)b + b^2) \]

### Step 5: Simplify the expression
Now simplify:
\[ = (a + 2b)((a + b)^2 - (a + b)b + b^2) \]
Calculating \( (a + b)^2 \):
\[ = a^2 + 2ab + b^2 \]
Now substituting back:
\[ = (a + 2b)(a^2 + 2ab + b^2 - ab - b^2) \]
This simplifies to:
\[ = (a + 2b)(a^2 + ab) \]

### Final Factorised Form
Thus, the final factorised form of the expression \( a^3 + 3a^2b + 3ab^2 + 2b^3 \) is:
\[ (a + 2b)(a^2 + ab) \]

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