## Solving the Algebraic Equation: (x+3)³ - x(3x+1)² + (2x+1)(4x² - 2x + 1) = 28

This article will guide you through the steps to solve the algebraic equation: (x+3)³ - x(3x+1)² + (2x+1)(4x² - 2x + 1) = 28.

### Expanding the Equation

The first step is to expand the equation by multiplying out the terms:

**(x+3)³**: This can be expanded using the binomial theorem or by multiplying (x+3) by itself three times.

(x+3)³ = (x+3)(x+3)(x+3) = x³ + 9x² + 27x + 27**x(3x+1)²**: This can be expanded by multiplying (3x+1) by itself and then multiplying the result by x. x(3x+1)² = x(9x² + 6x + 1) = 9x³ + 6x² + x**(2x+1)(4x² - 2x + 1)**: This is a product of a sum and difference of squares. (2x+1)(4x² - 2x + 1) = (2x)³ + 1³ = 8x³ + 1

Now, the equation becomes:

x³ + 9x² + 27x + 27 - (9x³ + 6x² + x) + 8x³ + 1 = 28

### Simplifying the Equation

Next, simplify the equation by combining like terms:

(x³ - 9x³ + 8x³) + (9x² - 6x²) + (27x - x) + (27 + 1 - 28) = 0

This simplifies to:

**0x³ + 3x² + 26x = 0**

### Solving for x

Finally, we have a simplified quadratic equation: 3x² + 26x = 0. To solve for x, we can factor out a common factor of x:

**x(3x + 26) = 0**

This gives us two possible solutions:

**x = 0****3x + 26 = 0 => x = -26/3**

Therefore, the solutions to the equation (x+3)³ - x(3x+1)² + (2x+1)(4x² - 2x + 1) = 28 are **x = 0** and **x = -26/3**.