Factorise: `25(x+y)^(2)- 36 (x-2y)^(2).`

Answer

To factorise the expression \( 25(x+y)^{2} - 36(x-2y)^{2} \), we can follow these steps:

### Step 1: Identify the squares
We can rewrite the expression as:
\[ (5(x+y))^2 - (6(x-2y))^2 \]
This is because \( 25 = 5^2 \) and \( 36 = 6^2 \).

### Step 2: Recognize the difference of squares
The expression now takes the form of a difference of squares, which is \( a^2 - b^2 \). We can use the identity:
\[ a^2 - b^2 = (a + b)(a - b) \]
Here, let:
- \( a = 5(x+y) \)
- \( b = 6(x-2y) \)

### Step 3: Apply the difference of squares formula
Now we can write:
\[ (5(x+y) + 6(x-2y))(5(x+y) - 6(x-2y)) \]

### Step 4: Simplify each part
Now we will simplify \( a + b \) and \( a - b \):

1. **For \( a + b \)**:
\[ 5(x+y) + 6(x-2y) = 5x + 5y + 6x - 12y = (5x + 6x) + (5y - 12y) = 11x - 7y \]

2. **For \( a - b \)**:
\[ 5(x+y) - 6(x-2y) = 5x + 5y - 6x + 12y = (5x - 6x) + (5y + 12y) = -x + 17y \]

### Step 5: Write the final factorised form
Putting it all together, we have:
\[ (11x - 7y)(-x + 17y) \]

### Final Answer:
Thus, the factorised form of \( 25(x+y)^{2} - 36(x-2y)^{2} \) is:
\[ (11x - 7y)(-x + 17y) \]