Factorise :
`(i) a^(3)+27b^(3)+8c^(3)-18abc " " (ii) 2sqrt(2)a^(3)+8b^(3)-27c^(3)+18sqrt(2)abc`
`(iii) x^(3)+y^(3)-12xy+64 " " (iv) 125-8x^(3)-27y^(3)=90xy`.

Answer

To factorize the given expressions, we will use the formula for the sum of cubes and the specific form of polynomials. The general formula for the sum of cubes is:

\[ a^3 + b^3 + c^3 - 3abc = (a + b + c)(a^2 + b^2 + c^2 - ab - ac - bc) \]

Let's apply this to each of the given expressions step by step.

### (i) Factorize \( a^3 + 27b^3 + 8c^3 - 18abc \)

1. Identify \( a, b, c \):
- \( a = a \)
- \( b = 3b \) (since \( 27b^3 = (3b)^3 \))
- \( c = 2c \) (since \( 8c^3 = (2c)^3 \))

2. Check if it fits the formula:
- We have \( 3abc = 3 \cdot a \cdot (3b) \cdot (2c) = 18abc \), which matches.

3. Apply the formula:
\[ a^3 + (3b)^3 + (2c)^3 - 3 \cdot a \cdot (3b) \cdot (2c) = (a + 3b + 2c)(a^2 + (3b)^2 + (2c)^2 - a(3b) - (3b)(2c) - (2c)(a)) \]

4. Simplify:
- \( a^2 + 9b^2 + 4c^2 - 3ab - 6bc - 2ca \)

5. Final factorized form:
\[ (a + 3b + 2c)(a^2 + 9b^2 + 4c^2 - 3ab - 6bc - 2ca) \]

### (ii) Factorize \( 2\sqrt{2}a^3 + 8b^3 - 27c^3 + 18\sqrt{2}abc \)

1. Identify \( a, b, c \):
- \( a = \sqrt{2}a \)
- \( b = 2b \)
- \( c = -3c \) (since \( -27c^3 = (-3c)^3 \))

2. Check if it fits the formula:
- \( 3abc = 3 \cdot (\sqrt{2}a) \cdot (2b) \cdot (-3c) = -18\sqrt{2}abc \), which matches.

3. Apply the formula:
\[ (\sqrt{2}a + 2b - 3c)(\sqrt{2}^2a^2 + (2b)^2 + (-3c)^2 - \sqrt{2}a(2b) - (2b)(-3c) - (-3c)(\sqrt{2}a)) \]

4. Simplify:
- \( 2a^2 + 4b^2 + 9c^2 - 2\sqrt{2}ab + 6bc + 3\sqrt{2}ac \)

5. Final factorized form:
\[ (\sqrt{2}a + 2b - 3c)(2a^2 + 4b^2 + 9c^2 - 2\sqrt{2}ab + 6bc + 3\sqrt{2}ac) \]

### (iii) Factorize \( x^3 + y^3 - 12xy + 64 \)

1. Identify \( a, b, c \):
- \( a = x \)
- \( b = y \)
- \( c = 4 \) (since \( 64 = 4^3 \))

2. Check if it fits the formula:
- \( 3abc = 3 \cdot x \cdot y \cdot 4 = 12xy \), which matches.

3. Apply the formula:
\[ (x + y + 4)(x^2 + y^2 + 4^2 - xy - 4y - 4x) \]

4. Simplify:
- \( x^2 + y^2 + 16 - xy - 4y - 4x \)

5. Final factorized form:
\[ (x + y + 4)(x^2 + y^2 + 16 - xy - 4y - 4x) \]

### (iv) Factorize \( 125 - 8x^3 - 27y^3 = 90xy \)

1. Rearrange:
\[ 125 - 8x^3 - 27y^3 - 90xy = 0 \]

2. Identify \( a, b, c \):
- \( a = 5 \) (since \( 125 = 5^3 \))
- \( b = -2x \)
- \( c = -3y \)

3. Check if it fits the formula:
- \( 3abc = 3 \cdot 5 \cdot (-2x) \cdot (-3y) = 30xy \), which does not match \( -90xy \).

4. Rewrite:
\[ 5^3 + (-2x)^3 + (-3y)^3 - 3 \cdot 5 \cdot (-2x) \cdot (-3y) = 0 \]

5. Apply the formula:
\[ (5 - 2x - 3y)(5^2 + (-2x)^2 + (-3y)^2 - 5(-2x) - (-2x)(-3y) - (-3y)(5)) \]

6. Simplify:
- \( 25 + 4x^2 + 9y^2 + 10x - 6xy + 15y \)

7. Final factorized form:
\[ (5 - 2x - 3y)(25 + 4x^2 + 9y^2 + 10x - 6xy + 15y) \]