1. **State the problem:** Divide the polynomial $2a^3 - a - 1$ by the binomial $2a + 3$. 2. **Formula and rules:** Polynomial division is similar to long division with numbers. We divide the highest degree term of the dividend by the highest degree term of the divisor, multiply the divisor by this result, subtract, and repeat until the remainder degree is less than the divisor degree. 3. **Step 1:** Divide the leading term $2a^3$ by $2a$: $$\frac{2a^3}{2a} = a^2$$ 4. **Step 2:** Multiply the divisor by $a^2$: $$a^2 \times (2a + 3) = 2a^3 + 3a^2$$ 5. **Step 3:** Subtract this from the original polynomial: $$\left(2a^3 - a - 1\right) - \left(2a^3 + 3a^2\right) = -3a^2 - a - 1$$ 6. **Step 4:** Divide the new leading term $-3a^2$ by $2a$: $$\frac{-3a^2}{2a} = -\frac{3}{2}a$$ 7. **Step 5:** Multiply the divisor by $-\frac{3}{2}a$: $$-\frac{3}{2}a \times (2a + 3) = -3a^2 - \frac{9}{2}a$$ 8. **Step 6:** Subtract this from the previous remainder: $$\left(-3a^2 - a - 1\right) - \left(-3a^2 - \frac{9}{2}a\right) = \cancel{-3a^2} - a - 1 - \left(\cancel{-3a^2} - \frac{9}{2}a\right) = \left(-a + \frac{9}{2}a\right) - 1 = \frac{7}{2}a - 1$$ 9. **Step 7:** Divide the leading term $\frac{7}{2}a$ by $2a$: $$\frac{\frac{7}{2}a}{2a} = \frac{7}{4}$$ 10. **Step 8:** Multiply the divisor by $\frac{7}{4}$: $$\frac{7}{4} \times (2a + 3) = \frac{7}{2}a + \frac{21}{4}$$ 11. **Step 9:** Subtract this from the previous remainder: $$\left(\frac{7}{2}a - 1\right) - \left(\frac{7}{2}a + \frac{21}{4}\right) = \cancel{\frac{7}{2}a} - 1 - \left(\cancel{\frac{7}{2}a} + \frac{21}{4}\right) = -1 - \frac{21}{4} = -\frac{25}{4}$$ 12. **Conclusion:** The quotient is $$Q(a) = a^2 - \frac{3}{2}a + \frac{7}{4}$$ and the remainder is $$R = -\frac{25}{4}$$ This means: $$2a^3 - a - 1 = (2a + 3)\left(a^2 - \frac{3}{2}a + \frac{7}{4}\right) - \frac{25}{4}$$