Fractorise:
`63a ^(2) - 112b^(2)`

Answer

To factorize the expression \( 63a^2 - 112b^2 \), we can follow these steps:

### Step 1: Identify the common factor
First, we look for the greatest common factor (GCF) of the coefficients 63 and 112. The GCF is 7.

### Step 2: Factor out the GCF
We can factor out 7 from the expression:
\[ 63a^2 - 112b^2 = 7(9a^2 - 16b^2) \]

### Step 3: Recognize the difference of squares
Next, we notice that \( 9a^2 - 16b^2 \) is a difference of squares. The difference of squares formula is:
\[ A^2 - B^2 = (A + B)(A - B) \]
In our case, \( A^2 = 9a^2 \) and \( B^2 = 16b^2 \).

### Step 4: Identify \( A \) and \( B \)
From the expressions:
- \( A = 3a \) (since \( (3a)^2 = 9a^2 \))
- \( B = 4b \) (since \( (4b)^2 = 16b^2 \))

### Step 5: Apply the difference of squares formula
Now we can apply the difference of squares formula:
\[ 9a^2 - 16b^2 = (3a + 4b)(3a - 4b) \]

### Step 6: Combine the factors
Putting it all together, we have:
\[ 63a^2 - 112b^2 = 7(3a + 4b)(3a - 4b) \]

### Final Answer
Thus, the factorized form of \( 63a^2 - 112b^2 \) is:
\[ 7(3a + 4b)(3a - 4b) \]

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