Factorise ` 16x^2+4y^2+9z^2-16xy-12yz+24xz`

Answer

To factorise the expression \( 16x^2 + 4y^2 + 9z^2 - 16xy - 12yz + 24xz \), we can follow these steps:

### Step 1: Rewrite the expression in a recognizable form
We can start by rewriting the squares and the cross terms in a more structured way.

1. **Identify squares**:
- \( 16x^2 = (4x)^2 \)
- \( 4y^2 = (2y)^2 \)
- \( 9z^2 = (3z)^2 \)

2. **Identify the cross terms**:
- The term \( -16xy \) can be rewritten as \( -2 \cdot (4x) \cdot (2y) \).
- The term \( -12yz \) can be rewritten as \( -2 \cdot (2y) \cdot (3z) \).
- The term \( 24xz \) can be rewritten as \( 2 \cdot (4x) \cdot (3z) \).

### Step 2: Group the terms
Now we can express the polynomial as follows:
\[ (4x)^2 + (2y)^2 + (3z)^2 - 2 \cdot (4x)(2y) - 2 \cdot (2y)(3z) + 2 \cdot (4x)(3z) \]

### Step 3: Recognize the pattern
This expression matches the form of the expansion of \( (A + B + C)^2 \):
\[ A^2 + B^2 + C^2 + 2AB + 2BC + 2CA \]
where:
- \( A = 4x \)
- \( B = -2y \)
- \( C = 3z \)

### Step 4: Write the factorization
Thus, we can factor the expression as:
\[ (4x - 2y + 3z)^2 \]

### Step 5: Final answer
Therefore, the factorized form of the expression \( 16x^2 + 4y^2 + 9z^2 - 16xy - 12yz + 24xz \) is:
\[ (4x - 2y + 3z)(4x - 2y + 3z) \quad \text{or} \quad (4x - 2y + 3z)^2 \]

---