Factorise:
`12x ^(2) y ^(3) - 21x^(3) y ^(2)`

Answer

To factorise the expression \( 12x^2y^3 - 21x^3y^2 \), we will follow these steps:

### Step 1: Identify the common factors
First, we need to identify the common factors in both terms of the expression. The coefficients are 12 and 21. The greatest common divisor (GCD) of 12 and 21 is 3.

Next, we look at the variables:
- For \( x^2 \) and \( x^3 \), the common factor is \( x^2 \).
- For \( y^3 \) and \( y^2 \), the common factor is \( y^2 \).

Thus, the common factor for the entire expression is \( 3x^2y^2 \).

### Step 2: Factor out the common factor
Now, we will factor out \( 3x^2y^2 \) from each term in the expression:

\[ 12x^2y^3 - 21x^3y^2 = 3x^2y^2(4y - 7x) \]

### Step 3: Write the final factored form
The final factored form of the expression is:

\[ 3x^2y^2(4y - 7x) \]

### Summary of Steps:
1. Identify the GCD of the coefficients and the common variables.
2. Factor out the common factor from the expression.
3. Write the expression in its factored form.

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