Factorise : (ii) `9(x+y)^2 - 16 (x-3y)^2`

Answer

To factorise the expression \( 9(x+y)^2 - 16(x-3y)^2 \), we can follow these steps:

### Step 1: Recognize the difference of squares
The expression is in the form \( a^2 - b^2 \), where:
- \( a = 3(x+y) \)
- \( b = 4(x-3y) \)

### Step 2: Apply the difference of squares formula
Using the difference of squares formula \( a^2 - b^2 = (a-b)(a+b) \):
\[ 9(x+y)^2 - 16(x-3y)^2 = (3(x+y) - 4(x-3y))(3(x+y) + 4(x-3y)) \]

### Step 3: Simplify each factor
Now we need to simplify both factors:

**For the first factor:**
\[ 3(x+y) - 4(x-3y) = 3x + 3y - 4x + 12y = -x + 15y = -(x - 15y) \]

**For the second factor:**
\[ 3(x+y) + 4(x-3y) = 3x + 3y + 4x - 12y = 7x - 9y \]

### Step 4: Combine the factors
Putting it all together, we have:
\[ 9(x+y)^2 - 16(x-3y)^2 = -(x - 15y)(7x - 9y) \]

### Final Answer
Thus, the factorised form of the expression is:
\[ -(x - 15y)(7x - 9y) \]

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