## Solving the Equation (x+3)(x^2-3x+9)-x(x-1)(x+1)=27

This article will guide you through the steps to solve the equation (x+3)(x^2-3x+9)-x(x-1)(x+1)=27.

### Understanding the Equation

The equation involves the product of two expressions:

**(x+3)(x^2-3x+9)**This is a special product known as the sum of cubes. It can be simplified using the formula: (a + b)(a^2 - ab + b^2) = a^3 + b^3.**x(x-1)(x+1)**This is the product of three consecutive integers. It can be simplified using the formula: x(x-1)(x+1) = x^3 - x.

### Step-by-Step Solution

**Expand the products:**- Using the sum of cubes formula, we get: (x+3)(x^2-3x+9) = x^3 + 3^3 = x^3 + 27
- Using the formula for the product of three consecutive integers, we get: x(x-1)(x+1) = x^3 - x

**Substitute the expanded expressions into the original equation:**- (x^3 + 27) - (x^3 - x) = 27

**Simplify the equation:**- x^3 + 27 - x^3 + x = 27
- x + 27 = 27

**Isolate x:**- x = 27 - 27
- x = 0

### Solution

Therefore, the solution to the equation (x+3)(x^2-3x+9)-x(x-1)(x+1)=27 is **x = 0**.

### Verification

To verify the solution, substitute x = 0 back into the original equation:

- (0+3)(0^2-3(0)+9) - 0(0-1)(0+1) = 27
- (3)(9) - 0 = 27
- 27 = 27

The equation holds true, confirming that x = 0 is indeed the solution.