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Let's work on factorizing the given expression:

\[
18y - 6x - 3(x - 3y)^2
\]

### Step 1: Expand \( -3(x - 3y)^2 \)

We need to first expand the quadratic term \( (x - 3y)^2 \).

\[
(x - 3y)^2 = (x - 3y)(x - 3y) = x^2 - 6xy + 9y^2
\]

Now, multiply the entire expression by \( -3 \):

\[
-3(x^2 - 6xy + 9y^2) = -3x^2 + 18xy - 27y^2
\]

### Step 2: Rewrite the whole expression

Now, substitute the expanded term back into the original equation:

\[
18y - 6x - 3(x^2 - 6xy + 9y^2) = 18y - 6x - 3x^2 + 18xy - 27y^2
\]

### Step 3: Rearrange terms

Let's rearrange the terms by grouping them together:

\[
-3x^2 + 18xy - 6x + 18y - 27y^2
\]

### Step 4: Factor grouping

We can now factor the expression by grouping terms. First, group the terms as follows:

\[
(-3x^2 - 6x) + (18xy + 18y) - 27y^2
\]

Now, factor each group:

\[
-3x(x + 2) + 18y(x + 1) - 27y^2
\]

At this stage, it seems there is no easy further factorization beyond grouping terms like this.

Would you like more details on any specific step or have any further questions?

Here are 5 related questions:
1. How do you fully factor trinomials like \( x^2 + bx + c \)?
2. Can we factor higher-degree polynomials using similar techniques?
3. What is the difference between factorizing and expanding expressions?
4. How does grouping work in factoring complex expressions?
5. What are common factoring methods for polynomials involving multiple variables?

**Tip:** When factorizing, always check if common factors can be factored out first!

Factorise expression 18y - 6x - 3(x - 3y)^2

Learn how to factorize the expression 18y - 6x - 3(x - 3y)^2 with detailed steps. The solution involves expanding the quadratic term and applying the distributive property for full factorization. Suitable for high school students in Grades 10-12.

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