Let's simplify the given expression:

$4y^3 + 2y^2 - 2y^2 - 2 - 3y - 3y$

Step-by-step simplification:

1. Combine like terms:
   • For $y^3$, there's only $4y^3$.
   • For $y^2$, we have $2y^2 - 2y^2$, which cancels out to 0.
   • For $y$, we have $-3y - 3y = -6y$.
   • The constant term is $-2$.

Thus, the simplified expression is:

$4y^3 - 6y - 2$

This matches the first option: $\boxed{4y^3 - 6y - 2}$

Would you like more details on any step or have any other questions?

Here are 5 related questions to deepen your understanding:

1. How do you combine like terms in polynomials?
2. Why did the $2y^2 - 2y^2$ terms cancel out?
3. How would you factor $4y^3 - 6y - 2$?
4. Can this expression be factored further to find roots?
5. What would happen if additional terms like $+5$ were included?

Tip: Always look for like terms first when simplifying algebraic expressions, as this often makes the problem much easier.