The expression \( n(n+1)(n+2) \), where \( n \in \mathbb{N} \) (the set of natural numbers), represents the product of three consecutive natural numbers: \( n \), \( n+1 \), and \( n+2 \).

To simplify or further expand this expression:

1. Multiply \( n \) with \( n+1 \):
   \[
   n(n+1) = n^2 + n
   \]

2. Now, multiply the result by \( n+2 \):
   \[
   (n^2 + n)(n+2) = n^2(n+2) + n(n+2)
   \]
   Expanding both terms:
   \[
   = n^3 + 2n^2 + n^2 + 2n
   \]

3. Combine like terms:
   \[
   n^3 + 3n^2 + 2n
   \]

Thus, the expanded form of \( n(n+1)(n+2) \) is:
\[
n^3 + 3n^2 + 2n
\]

### Do you have any questions or would you like more details?

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Here are 5 related questions to explore further:

1. How does the expansion change if \( n(n+1)(n+2)(n+3) \) is expressed?
2. Can you find the sum of the first \( n \) natural numbers?
3. What is the product of four consecutive natural numbers, \( n(n+1)(n+2)(n+3) \)?
4. How do you factor the cubic expression \( n^3 + 3n^2 + 2n \)?
5. What is the value of \( n(n+1)(n+2) \) for \( n = 4 \)?

**Tip:** When expanding polynomials, always carefully distribute each term to avoid errors.

Explore how to expand and simplify the expression n(n+1)(n+2), where n is a natural number. This problem covers key algebraic concepts and polynomial expansions, suitable for students in Grades 8-10.

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