factorize : `8ab^2-18a^3`

Answer

To factorize the expression \(8ab^2 - 18a^3\), we will follow these steps:

### Step 1: Identify the common factors
The first step is to identify the common factors in both terms of the expression. The terms are \(8ab^2\) and \(-18a^3\).

**Common factors**:
- The coefficients are \(8\) and \(18\). The greatest common divisor (GCD) of \(8\) and \(18\) is \(2\).
- Both terms have \(a\) as a common factor.

Thus, the common factor is \(2a\).

### Step 2: Factor out the common factor
Now, we will factor out \(2a\) from the expression:

\[ 8ab^2 - 18a^3 = 2a(4b^2 - 9a^2) \]

### Step 3: Recognize the difference of squares
Next, we observe that the expression inside the parentheses, \(4b^2 - 9a^2\), is a difference of squares. It can be expressed as:

\[ 4b^2 - 9a^2 = (2b)^2 - (3a)^2 \]

### Step 4: Apply the difference of squares formula
The difference of squares can be factored using the formula \(x^2 - y^2 = (x - y)(x + y)\). Here, \(x = 2b\) and \(y = 3a\):

\[ (2b)^2 - (3a)^2 = (2b - 3a)(2b + 3a) \]

### Step 5: Combine the factors
Now we can combine everything back together:

\[ 8ab^2 - 18a^3 = 2a(2b - 3a)(2b + 3a) \]

### Final Answer
Thus, the completely factored form of the expression \(8ab^2 - 18a^3\) is:

\[ \boxed{2a(2b - 3a)(2b + 3a)} \]

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