5. Factorise: (i) 4x 2 + 9y 2 + 16z 2 + 12xy – 24yz – 16xz   
can you tell me how to do this

Sure! Let's factor the expression step by step. The expression we are dealing with is:

\[ 4x^2 + 9y^2 + 16z^2 + 12xy - 24yz - 16xz \]

### Step 1: Rearrange the terms
Group the terms based on similar pairs:

\[
4x^2 + 9y^2 + 12xy + 16z^2 - 16xz - 24yz
\]

### Step 2: Regroup the expression into pairs
Now let's regroup and factor by pairs:

\[
(4x^2 + 12xy + 9y^2) + (16z^2 - 16xz - 24yz)
\]

### Step 3: Factor each group separately
#### For the first group: \(4x^2 + 12xy + 9y^2\)
This is a perfect square trinomial. It can be factored as:

\[
(2x + 3y)^2
\]

#### For the second group: \(16z^2 - 16xz - 24yz\)
We can factor out a common factor of 4:

\[
4(4z^2 - 4xz - 6yz)
\]

Now, factor the quadratic inside the parentheses:

\[
4(2z - x - 3y)^2
\]

### Step 4: Write the factored form
Putting both parts together, the factored form of the original expression is:

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

### Step 5: Apply the difference of squares
This is now a difference of squares, so we can apply the difference of squares formula:

\[
(a^2 - b^2 = (a - b)(a + b))
\]

Thus, we get:

\[
\left((2x + 3y) - 2(2z - x - 3y)\right)\left((2x + 3y) + 2(2z - x - 3y)\right)
\]

Now, expand each factor:

\[
(2x + 3y - 4z + 2x + 6y) \quad \text{and} \quad (2x + 3y + 4z - 2x - 6y)
\]

Simplifying the two factors:

\[
(4x + 9y - 4z)(4z - 3y)
\]

So the completely factored form of the expression is:

\[
(4x + 9y - 4z)(4z - 3y)
\]

This is the final factorization.

---

Would you like a more detailed breakdown of any part of this process?

Here are 5 related questions to practice:
1. Factorize \( x^2 + 6x + 9 \).
2. Factor the expression \( 9a^2 - 25b^2 \).
3. Simplify and factor \( 3x^2 - 12x + 12 \).
4. What is the factored form of \( 16x^2 - 24xy + 9y^2 \)?
5. Factor the expression \( x^2 - 5x + 6 \).

**Tip:** When dealing with quadratic trinomials, always check if it's a perfect square or can be simplified using common factorization tricks.

Factorise: (i) 4x^2 + 9y^2 + 16z^2 + 12xy – 24yz – 16xz. Can you tell me how to do this?

Discover how to factorize the complex algebraic expression 4x^2 + 9y^2 + 16z^2 + 12xy – 24yz – 16xz using perfect square trinomials and the difference of squares. This step-by-step guide is ideal for students in Grades 10-12 learning advanced algebraic techniques.

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