Find the product :
`(i) (a+2b+4c)(a^(2)+4b^(2)+16c^(2)-2ab-8bc-4ca)`
`(ii) (3x-5y-4)(9x^(2)+25y^(2)+15xy+12x-20y+16)`
`(iii) (2-3b-7c)(4+9b^(2)+49c^(2)+6b-21bc+14c)`
`(iv) (sqrt(2)a+2sqrt(2)b+c)(2a^(2)+8b^(2)+c^(2)-4ab-2sqrt(2)bc-sqrt(2)ac)`

Answer

To find the products of the given expressions, we will use the algebraic identities and properties of polynomials. Let's solve each part step by step.

### (i) Find the product: \((a + 2b + 4c)(a^2 + 4b^2 + 16c^2 - 2ab - 8bc - 4ca)\)

1. **Distribute each term in the first polynomial to each term in the second polynomial**:
- \( a \cdot (a^2 + 4b^2 + 16c^2 - 2ab - 8bc - 4ca) \)
- \( 2b \cdot (a^2 + 4b^2 + 16c^2 - 2ab - 8bc - 4ca) \)
- \( 4c \cdot (a^2 + 4b^2 + 16c^2 - 2ab - 8bc - 4ca) \)

2. **Calculate each distribution**:
- \( a^3 + 4ab^2 + 16ac^2 - 2a^2b - 8abc - 4a^2c \)
- \( 2ba^2 + 8b^3 + 32bc^2 - 4b^2a - 16b^2c - 8bc^2 \)
- \( 4ca^2 + 16cb^2 + 64c^3 - 8cab - 32cbc - 16c^2a \)

3. **Combine like terms**:
- Combine all the terms from the distributions above.

4. **Final expression**:
- The final expression will be the sum of all the terms collected.