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The given expression to factorise is:

\[
3x^3y - 12x^2yz + 12xyz^2
\]

### Step 1: Identify the common factor

Each term has a common factor of \( 3xy \). Let's factor that out:

\[
3xy(x^2 - 4xz + 4z^2)
\]

### Step 2: Factor the quadratic inside the parentheses

The quadratic expression \( x^2 - 4xz + 4z^2 \) is a perfect square trinomial. It can be rewritten as:

\[
x^2 - 4xz + 4z^2 = (x - 2z)^2
\]

### Step 3: Write the final factorised form

Now, substitute this back into the expression:

\[
3xy(x - 2z)^2
\]

So, the completely factorised form is:

\[
\boxed{3xy(x - 2z)^2}
\]

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Would you like more details on any step or clarification?

Here are five related questions:
1. What is the process to identify common factors in polynomials?
2. How do you recognize perfect square trinomials?
3. Can all quadratic expressions be factorised into squares?
4. What happens if a term has no common factors?
5. How do we verify that a factorisation is correct?

**Tip:** Always check for common factors first before trying other factorisation techniques, as it simplifies the process.

This page provides a step-by-step guide to factorising the polynomial expression 3x^3y - 12x^2yz + 12xyz^2. Learn how to factor out the common terms and recognize perfect square trinomials for complete factorization. Suitable for students in Grades 9-11.

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