Factorise :
`9(a-2)^(2) - 16(a+2)^(2)`

Answer

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

### Step 1: Recognize the difference of squares
The expression \( 9(a-2)^2 - 16(a+2)^2 \) can be recognized as a difference of squares, which has the form \( A^2 - B^2 \). Here, we can let:
- \( A = 3(a-2) \)
- \( B = 4(a+2) \)

### Step 2: Apply the difference of squares formula
The difference of squares can be factored using the formula:
\[ A^2 - B^2 = (A + B)(A - B) \]
Substituting \( A \) and \( B \) into this formula gives:
\[ (3(a-2) + 4(a+2))(3(a-2) - 4(a+2)) \]

### Step 3: Simplify each factor
Now, we simplify each factor:
1. **For \( A + B \)**:
\[ 3(a-2) + 4(a+2) = 3a - 6 + 4a + 8 = 7a + 2 \]

2. **For \( A - B \)**:
\[ 3(a-2) - 4(a+2) = 3a - 6 - 4a - 8 = -a - 14 \]

### Step 4: Combine the factors
Now we can write the factorised form:
\[ (7a + 2)(-a - 14) \]

### Step 5: Factor out the negative sign
To make it look more standard, we can factor out the negative sign from the second factor:
\[ -(7a + 2)(a + 14) \]

### Final Answer
Thus, the factorised form of the expression \( 9(a-2)^2 - 16(a+2)^2 \) is:
\[ -(7a + 2)(a + 14) \]

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