3x(3x-1) - 2(3x-1)^2 - (3x-2)^2 - 3x(2-3x)
do me factorization with detail steps

To factor the expression \( 3x(3x - 1) - 2(3x - 1)^2 - (3x - 2)^2 - 3x(2 - 3x) \), let’s follow these steps carefully.

### Step 1: Expand all terms
We will expand each term individually.

#### First term: \( 3x(3x - 1) \)
\[
3x(3x - 1) = 3x \cdot 3x - 3x \cdot 1 = 9x^2 - 3x
\]

#### Second term: \( -2(3x - 1)^2 \)
We first expand \( (3x - 1)^2 \):
\[
(3x - 1)^2 = (3x - 1)(3x - 1) = 9x^2 - 6x + 1
\]
Now multiply by \( -2 \):
\[
-2(3x - 1)^2 = -2(9x^2 - 6x + 1) = -18x^2 + 12x - 2
\]

#### Third term: \( -(3x - 2)^2 \)
\[
(3x - 2)^2 = (3x - 2)(3x - 2) = 9x^2 - 12x + 4
\]
Now apply the negative sign:
\[
-(3x - 2)^2 = -(9x^2 - 12x + 4) = -9x^2 + 12x - 4
\]

#### Fourth term: \( -3x(2 - 3x) \)
\[
3x(2 - 3x) = 3x \cdot 2 - 3x \cdot 3x = 6x - 9x^2
\]
Apply the negative sign:
\[
-3x(2 - 3x) = -6x + 9x^2
\]

### Step 2: Combine all expanded terms
Now combine the terms from Step 1:
\[
9x^2 - 3x - 18x^2 + 12x - 2 - 9x^2 + 12x - 4 + 9x^2 - 6x
\]

Combine like terms:
- \( 9x^2 - 18x^2 - 9x^2 + 9x^2 = -9x^2 \)
- \( -3x + 12x + 12x - 6x - 6x = 9x \)
- Constant terms: \( -2 - 4 = -6 \)

So, the expression simplifies to:
\[
-9x^2 + 9x - 6
\]

### Step 3: Factor the simplified expression
Now we have the quadratic expression \( -9x^2 + 9x - 6 \). We can factor out a common factor of \( -3 \):
\[
-3(3x^2 - 3x + 2)
\]

At this point, the quadratic \( 3x^2 - 3x + 2 \) cannot be factored further using real numbers because its discriminant \( (-3)^2 - 4(3)(2) = 9 - 24 = -15 \) is negative, meaning it has complex roots.

Thus, the fully factored form is:
\[
-3(3x^2 - 3x + 2)
\]

### Final Answer:
The factorized form of \( 3x(3x - 1) - 2(3x - 1)^2 - (3x - 2)^2 - 3x(2 - 3x) \) is:
\[
-3(3x^2 - 3x + 2)
\]

Would you like more details or have any questions?

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**Related questions:**
1. How do you solve quadratics with complex roots?
2. What are the methods to factor quadratics?
3. How do discriminants help in solving quadratic equations?
4. What are common factoring techniques in algebra?
5. How can quadratic expressions be used in real-world problems?

**Tip:** Always check the discriminant when factoring quadratic expressions to determine if real or complex solutions exist.

3x(3x-1) - 2(3x-1)^2 - (3x-2)^2 - 3x(2-3x) do me factorization with detail steps

Explore the detailed factorization of the expression 3x(3x-1) - 2(3x-1)^2 - (3x-2)^2 - 3x(2-3x). This step-by-step solution covers key algebraic concepts such as binomial expansion, quadratic equations, and the use of the discriminant to find the factored form of the equation. Suitable for high school students, Grades 9-12.

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