1. **State the problem:** Divide the polynomial $8x^2 - 45y^2 + 18xy$ by the binomial $2x - 3y$. 2. **Formula and approach:** Polynomial division can be done using long division or synthetic division. Here, we use long division for polynomials in two variables. 3. **Arrange terms:** Write the dividend and divisor in standard form: Dividend: $8x^2 + 18xy - 45y^2$ Divisor: $2x - 3y$ 4. **Divide the leading term:** Divide the leading term of the dividend $8x^2$ by the leading term of the divisor $2x$: $$\frac{8x^2}{2x} = 4x$$ 5. **Multiply and subtract:** Multiply the divisor by $4x$: $$4x(2x - 3y) = 8x^2 - 12xy$$ Subtract this from the dividend: $$(8x^2 + 18xy - 45y^2) - (8x^2 - 12xy) = 0 + 30xy - 45y^2$$ 6. **Repeat division:** Divide the new leading term $30xy$ by $2x$: $$\frac{30xy}{2x} = 15y$$ 7. **Multiply and subtract:** Multiply the divisor by $15y$: $$15y(2x - 3y) = 30xy - 45y^2$$ Subtract this from the current remainder: $$(30xy - 45y^2) - (30xy - 45y^2) = 0$$ 8. **Conclusion:** The remainder is zero, so the division is exact. **Final answer:** $$\frac{8x^2 - 45y^2 + 18xy}{2x - 3y} = 4x + 15y$$