Multiplying Binomials: (7u^2+8uv6v^2)(6u^2+4uv+3v^2)
This problem involves multiplying two trinomials. To do this, we'll use the distributive property (often referred to as FOIL for binomials) multiple times.
Here's the breakdown:

Distribute the first term of the first trinomial:
 (7u^2)(6u^2+4uv+3v^2) = 42u^4 + 28u^3v + 21u^2v^2

Distribute the second term of the first trinomial:
 (8uv)(6u^2+4uv+3v^2) = 48u^3v + 32u^2v^2 + 24uv^3

Distribute the third term of the first trinomial:
 (6v^2)(6u^2+4uv+3v^2) = 36u^2v^2  24uv^3  18v^4

Combine like terms:
 42u^4 + 28u^3v + 21u^2v^2 + 48u^3v + 32u^2v^2 + 24uv^3  36u^2v^2  24uv^3  18v^4

Simplify:
 42u^4 + 76u^3v + 17u^2v^2  18v^4
Therefore, the product of (7u^2+8uv6v^2)(6u^2+4uv+3v^2) is 42u^4 + 76u^3v + 17u^2v^2  18v^4.
Key points to remember:
 FOIL (First, Outer, Inner, Last) is a helpful mnemonic for multiplying binomials, but it doesn't apply directly to trinomials.
 Organization: Write out each step clearly, making sure to distribute correctly and combine like terms.
 Practice: Practice multiplying binomials and trinomials to become more familiar with the process.