Factorise :
`8a^(3) - b^(3) - 4ax + 2bx`

Answer

To factorise the expression \(8a^3 - b^3 - 4ax + 2bx\), we can follow these steps:

### Step 1: Identify the terms
The expression is \(8a^3 - b^3 - 4ax + 2bx\). We can group the terms in a way that allows us to use the difference of cubes and factor out common factors.

### Step 2: Rearrange the expression
Rearranging the expression gives us:
\[ 8a^3 - b^3 + (-4ax + 2bx) \]
This can be written as:
\[ 8a^3 - b^3 - 4ax + 2bx \]

### Step 3: Factor the difference of cubes
We can use the identity for the difference of cubes:
\[ c^3 - d^3 = (c - d)(c^2 + d^2 + cd) \]
Here, let \(c = 2a\) and \(d = b\). Then, we have:
\[ 8a^3 - b^3 = (2a)^3 - b^3 \]
Applying the identity:
\[ = (2a - b)( (2a)^2 + b^2 + 2ab) \]
Calculating \( (2a)^2 + b^2 + 2ab \):
\[ = 4a^2 + b^2 + 2ab \]
So we have:
\[ 8a^3 - b^3 = (2a - b)(4a^2 + b^2 + 2ab) \]

### Step 4: Factor out common terms from the remaining expression
Now, we look at the remaining terms \(-4ax + 2bx\):
\[ -4ax + 2bx = -2x(2a - b) \]

### Step 5: Combine the factors
Now we can combine our results:
\[ 8a^3 - b^3 - 4ax + 2bx = (2a - b)(4a^2 + b^2 + 2ab) - 2x(2a - b) \]
Factoring out the common term \((2a - b)\):
\[ = (2a - b)(4a^2 + b^2 + 2ab - 2x) \]

### Final Factorised Form
Thus, the factorised form of the expression \(8a^3 - b^3 - 4ax + 2bx\) is:
\[ (2a - b)(4a^2 + b^2 + 2ab - 2x) \]

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