Factorise each of the following: (i) `8a^3+b^3+12 a^2b+6a b^2`
(ii) `8a^3-b^3-12 a^2b+6a b^2`(iii) `27-125 b^3-135 a+225 a^2`
(iv) `64 a^3-27 a^3-144 a^2b+108 a b^2`(v) `27 p^3-1/(216)-9/2p^2+1/4p`

Explanation

Let's solve the given questions step by step.

### (i) Factorise `8a^3 + b^3 + 12a^2b + 6ab^2`

1. **Identify the form**: Recognize that the expression can be related to the identity \(x^3 + y^3 + 3x^2y + 3xy^2 = (x + y)^3\).
2. **Rewrite the terms**:
- \(8a^3\) can be rewritten as \((2a)^3\).
- The other terms \(12a^2b\) and \(6ab^2\) can be expressed as \(3(2a)^2b\) and \(3(2a)(b^2)\).
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