## Solving the Equation: (x-3)^3 - (x-3)(x^2+3x+9) + 6(x+1)^2 + 3x^2 = -33

This equation appears complex, but we can simplify it using algebraic properties and then solve for the unknown variable 'x'. Let's break down the process:

### 1. Expanding the Expressions:

**(x-3)^3**: This is a cube of a binomial, which can be expanded using the formula: (a-b)^3 = a^3 - 3a^2b + 3ab^2 - b^3- Expanding, we get: x^3 - 9x^2 + 27x - 27

**(x-3)(x^2+3x+9)**: This is a product of a binomial and a trinomial, which is a special case of the sum of cubes: (a-b)(a^2+ab+b^2) = a^3 - b^3- Expanding, we get: x^3 - 27

**6(x+1)^2**: This is a product of a constant and a squared binomial, which can be expanded using the formula: (a+b)^2 = a^2 + 2ab + b^2- Expanding, we get: 6(x^2 + 2x + 1) = 6x^2 + 12x + 6

### 2. Substituting and Simplifying:

Now we can substitute the expanded expressions back into the original equation: (x^3 - 9x^2 + 27x - 27) - (x^3 - 27) + (6x^2 + 12x + 6) + 3x^2 = -33

Simplifying by combining like terms:

-9x^2 + 27x + 6x^2 + 12x + 3x^2 + 27 + 6 = -33 -9x^2 + 6x^2 + 3x^2 + 27x + 12x + 27 + 6 = -33 39x + 33 = -33

### 3. Isolating and Solving for 'x':

Subtracting 33 from both sides:

39x = -66

Dividing both sides by 39:

x = -66/39

Simplifying the fraction:

x = -22/13

### Solution:

Therefore, the solution to the equation (x-3)^3 - (x-3)(x^2+3x+9) + 6(x+1)^2 + 3x^2 = -33 is **x = -22/13**.