Factorise by taking out the common factors :
`xy(3x^(2) - 2y^(2)) - yz(2y^(2) - 3x^(2)) + zx(15x^(2) - 10y^(2))`

Answer

To factorise the expression \( xy(3x^{2} - 2y^{2}) - yz(2y^{2} - 3x^{2}) + zx(15x^{2} - 10y^{2}) \), we will follow these steps:

### Step 1: Rewrite the expression
We start with the expression:
\[ xy(3x^{2} - 2y^{2}) - yz(2y^{2} - 3x^{2}) + zx(15x^{2} - 10y^{2}) \]

### Step 2: Rearranging terms
Notice that we can rearrange the second term:
\[ - yz(2y^{2} - 3x^{2}) = - yz(-3x^{2} + 2y^{2}) = yz(3x^{2} - 2y^{2}) \]
This allows us to write the expression as:
\[ xy(3x^{2} - 2y^{2}) + yz(3x^{2} - 2y^{2}) + zx(15x^{2} - 10y^{2}) \]

### Step 3: Factor out the common terms
Now, we can see that the first two terms share a common factor of \( (3x^{2} - 2y^{2}) \):
\[ (3x^{2} - 2y^{2})(xy + yz) + zx(15x^{2} - 10y^{2}) \]

### Step 4: Factor out common factors from the third term
The third term can be simplified:
\[ zx(15x^{2} - 10y^{2}) = zx \cdot 5(3x^{2} - 2y^{2}) \]
So we can rewrite the expression as:
\[ (3x^{2} - 2y^{2})(xy + yz + 5zx) \]

### Final Result
Thus, the fully factorised form of the expression is:
\[ (3x^{2} - 2y^{2})(xy + yz + 5zx) \]