1. **State the problem:** We need to verify and understand the equation $$n^3 + 2 = n(n + 1)(n - 1) + 2$$ where $n \in \mathbb{N}$. 2. **Recall the formula:** The expression $n(n+1)(n-1)$ represents the product of three consecutive integers centered at $n$. This can be simplified using algebraic identities. 3. **Simplify the right-hand side:** $$n(n+1)(n-1) = n \times (n^2 - 1) = n^3 - n$$ 4. **Rewrite the original equation using this simplification:** $$n^3 + 2 = (n^3 - n) + 2$$ 5. **Subtract $n^3 + 2$ from both sides to check equality:** $$n^3 + 2 - (n^3 - n + 2) = 0$$ $$n^3 + 2 - n^3 + n - 2 = 0$$ $$n = 0$$ 6. **Interpretation:** The equation holds true if and only if $n = 0$. However, since $n \in \mathbb{N}$ (natural numbers), and typically $\mathbb{N} = \{1, 2, 3, ...\}$, the equation is not true for any natural number. **Final answer:** The equation $$n^3 + 2 = n(n + 1)(n - 1) + 2$$ is false for all natural numbers $n$ except possibly $n=0$ which is not in $\mathbb{N}$ under the usual definition.