## Solving the Equation: (x+1)^3 - (x-1)(x^2+x+1) - 2 = 0

This article aims to solve the equation (x+1)^3 - (x-1)(x^2+x+1) - 2 = 0. We will use algebraic manipulation and the properties of exponents and multiplication to simplify the equation and find its solution(s).

### Expanding the Equation

First, we expand the equation using the following rules:

**(a + b)^3 = a^3 + 3a^2b + 3ab^2 + b^3****(a - b)(a^2 + ab + b^2) = a^3 - b^3**

Applying these rules, we get:

(x+1)^3 - (x-1)(x^2+x+1) - 2 = 0 => x^3 + 3x^2 + 3x + 1 - (x^3 - 1) - 2 = 0

### Simplifying the Equation

Now, we simplify the equation by combining like terms:

x^3 + 3x^2 + 3x + 1 - x^3 + 1 - 2 = 0
=> **3x^2 + 3x = 0**

### Solving for x

To solve for x, we factor out a 3x from the equation:

**3x(x + 1) = 0**

This gives us two possible solutions:

**3x = 0**=>**x = 0****x + 1 = 0**=>**x = -1**

### Conclusion

Therefore, the solutions to the equation (x+1)^3 - (x-1)(x^2+x+1) - 2 = 0 are **x = 0** and **x = -1**.