Facorise: `27a^3-b^3+8c^3+18abc`

Answer

To factorise the expression \(27a^3 - b^3 + 8c^3 + 18abc\), we can follow these steps:

### Step 1: Rearrange the expression
First, let's rearrange the expression to group similar terms:
\[ 27a^3 + 18abc + 8c^3 - b^3 \]

### Step 2: Identify the terms
We can identify the terms as follows:
- \(A = 27a^3\)
- \(B = -b^3\)
- \(C = 8c^3\)
- \(D = 18abc\)

### Step 3: Recognize the form
Notice that the expression resembles the form of a polynomial that can be factored using the identity for the sum of cubes and the product of three terms:
\[ A + B + C - 3abc \]
where \(A = (3a)^3\), \(B = (-b)^3\), and \(C = (2c)^3\).

### Step 4: Apply the factorization formula
Using the identity:
\[ x^3 + y^3 + z^3 - 3xyz = (x + y + z)(x^2 + y^2 + z^2 - xy - xz - yz) \]
we can substitute:
- \(x = 3a\)
- \(y = -b\)
- \(z = 2c\)

### Step 5: Calculate \(x + y + z\)
Calculate:
\[ x + y + z = 3a - b + 2c \]

### Step 6: Calculate \(x^2 + y^2 + z^2 - xy - xz - yz\)
Now calculate:
\[ x^2 = (3a)^2 = 9a^2 \]
\[ y^2 = (-b)^2 = b^2 \]
\[ z^2 = (2c)^2 = 4c^2 \]
\[ xy = (3a)(-b) = -3ab \]
\[ xz = (3a)(2c) = 6ac \]
\[ yz = (-b)(2c) = -2bc \]

Now combine these:
\[ x^2 + y^2 + z^2 - xy - xz - yz = 9a^2 + b^2 + 4c^2 - (-3ab) - 6ac - (-2bc) \]
This simplifies to:
\[ 9a^2 + b^2 + 4c^2 + 3ab - 6ac + 2bc \]

### Step 7: Write the final factorization
Thus, the complete factorization of the expression is:
\[ (3a - b + 2c)(9a^2 + b^2 + 4c^2 + 3ab - 6ac + 2bc) \]

### Final Answer
The factorized form of \(27a^3 - b^3 + 8c^3 + 18abc\) is:
\[ (3a - b + 2c)(9a^2 + b^2 + 4c^2 + 3ab - 6ac + 2bc) \]