factorize : `20x^2-45`

Answer

To factorize the expression \(20x^2 - 45\), we can follow these steps:

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

### Step 2: Factor out the GCF
Now, we can factor out \(5\) from the expression:
\[ 20x^2 - 45 = 5(4x^2 - 9) \]

### Step 3: Recognize the difference of squares
Next, we notice that \(4x^2 - 9\) can be expressed as a difference of squares. Recall that \(a^2 - b^2 = (a + b)(a - b)\). Here, \(4x^2\) is \((2x)^2\) and \(9\) is \(3^2\).

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

### Step 5: Write the final factorization
Now, substituting back into our expression, we have:
\[ 20x^2 - 45 = 5(2x + 3)(2x - 3) \]

Thus, the factorization of \(20x^2 - 45\) is:
\[ \boxed{5(2x + 3)(2x - 3)} \]

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