Divide : `8x^(2) - 45y^(2) + 18xy` by 2x - 3y.

Answer

To divide the polynomial \(8x^2 - 45y^2 + 18xy\) by \(2x - 3y\) using long division, we will follow these steps:

### Step 1: Set up the division
We need to divide \(8x^2 - 45y^2 + 18xy\) by \(2x - 3y\). We will write it in long division format.

### Step 2: Divide the leading term
We take the leading term of the dividend \(8x^2\) and divide it by the leading term of the divisor \(2x\):
\[ \frac{8x^2}{2x} = 4x \]
This means our first term in the quotient is \(4x\).

### Step 3: Multiply and subtract
Next, we multiply the entire divisor \(2x - 3y\) by \(4x\):
\[ 4x \cdot (2x - 3y) = 8x^2 - 12xy \]
Now, we subtract this result from the original polynomial:
\[ (8x^2 - 45y^2 + 18xy) - (8x^2 - 12xy) = -45y^2 + 18xy + 12xy = -45y^2 + 30xy \]

### Step 4: Repeat the process
Now we need to divide the new leading term \(-45y^2\) by the leading term of the divisor \(2x\):
\[ \frac{-45y^2}{2x} = -\frac{45}{2} \cdot \frac{y^2}{x} \]
However, since we are looking for a term that can be multiplied by the divisor to cancel out the \(-45y^2\), we will instead focus on the next term.

### Step 5: Divide the leading term
Now, we will consider the term \(-45y^2\) and divide it by the leading term of the divisor \(-3y\):
\[ \frac{-45y^2}{-3y} = 15y \]
So, we add \(15y\) to our quotient.

### Step 6: Multiply and subtract again
Next, we multiply the entire divisor \(2x - 3y\) by \(15y\):
\[ 15y \cdot (2x - 3y) = 30xy - 45y^2 \]
Now we subtract this from our current polynomial:
\[ (-45y^2 + 30xy) - (30xy - 45y^2) = 0 \]

### Final Result
Since the remainder is \(0\), we conclude that the division is exact. Therefore, the quotient is:
\[ \text{Quotient} = 4x + 15y \]

### Summary
The result of dividing \(8x^2 - 45y^2 + 18xy\) by \(2x - 3y\) is:
\[ \boxed{4x + 15y} \]