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

Answer

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

### Step 1: Identify the structure of the polynomial
We start by rewriting the polynomial in a more recognizable form. Notice that it consists of squares and products of variables.

### Step 2: Group the terms
We can rearrange the polynomial as follows:
\[ 9x^2 - 24xy + 16y^2 + 4z^2 - 12xz + 16yz \]
This helps us see the quadratic nature of the terms involving \(x\), \(y\), and \(z\).

### Step 3: Rewrite the quadratic terms
We can express the first three terms \(9x^2 - 24xy + 16y^2\) as a perfect square:
\[ 9x^2 - 24xy + 16y^2 = (3x - 4y)^2 \]

### Step 4: Rewrite the remaining terms
Now, we look at the remaining terms \(4z^2 - 12xz + 16yz\). We can express this as:
\[ 4z^2 - 12xz + 16yz = (2z - 3x + 4y)^2 \]

### Step 5: Combine the factored forms
Now we can combine the factored forms:
\[ (3x - 4y - 2z)^2 \]

### Final Step: Write the final factorized form
Thus, the complete factorization of the polynomial is:
\[ (3x - 4y - 2z)^2 \]