Fractorise:
`20a ^(2) - 45 b ^(2)`

Answer

To factorize the expression \( 20a^2 - 45b^2 \), we will follow these steps:

### Step 1: Identify the common factor
First, we need to find the greatest common factor (GCF) of the coefficients 20 and 45. The GCF of 20 and 45 is 5.

### Step 2: Factor out the common factor
Next, we factor out the GCF from the expression:
\[ 20a^2 - 45b^2 = 5(4a^2 - 9b^2) \]

### Step 3: Recognize the difference of squares
Now, we observe that \( 4a^2 - 9b^2 \) is a difference of squares. The difference of squares formula is:
\[ x^2 - y^2 = (x + y)(x - y) \]
In our case, we can identify:
- \( x^2 = 4a^2 \) which means \( x = 2a \)
- \( y^2 = 9b^2 \) which means \( y = 3b \)

### Step 4: Apply the difference of squares formula
Using the difference of squares formula, we can rewrite \( 4a^2 - 9b^2 \):
\[ 4a^2 - 9b^2 = (2a + 3b)(2a - 3b) \]

### Step 5: Combine the factors
Now, we combine the factors we found:
\[ 20a^2 - 45b^2 = 5(2a + 3b)(2a - 3b) \]

### Final Answer
Thus, the factorized form of \( 20a^2 - 45b^2 \) is:
\[ 5(2a + 3b)(2a - 3b) \]

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