1. **State the problem:** Divide the polynomial $4x^3 + x + 1$ by the binomial $2x + 3$. 2. **Formula and rules:** Polynomial division is similar to long division with numbers. We divide the leading term of the dividend by the leading term of the divisor, multiply the divisor by that result, subtract, and repeat until the degree of the remainder is less than the divisor. 3. **Step 1:** Divide the leading term $4x^3$ by $2x$: $$\frac{4x^3}{2x} = 2x^2$$ 4. **Step 2:** Multiply the divisor by $2x^2$: $$2x^2 \times (2x + 3) = 4x^3 + 6x^2$$ 5. **Step 3:** Subtract this from the original polynomial: $$\left(4x^3 + 0x^2 + x + 1\right) - \left(4x^3 + 6x^2\right) = 0x^3 - 6x^2 + x + 1 = -6x^2 + x + 1$$ 6. **Step 4:** Divide the new leading term $-6x^2$ by $2x$: $$\frac{-6x^2}{2x} = -3x$$ 7. **Step 5:** Multiply the divisor by $-3x$: $$-3x \times (2x + 3) = -6x^2 - 9x$$ 8. **Step 6:** Subtract: $$(-6x^2 + x + 1) - (-6x^2 - 9x) = 0x^2 + 10x + 1 = 10x + 1$$ 9. **Step 7:** Divide $10x$ by $2x$: $$\frac{10x}{2x} = 5$$ 10. **Step 8:** Multiply divisor by $5$: $$5 \times (2x + 3) = 10x + 15$$ 11. **Step 9:** Subtract: $$ (10x + 1) - (10x + 15) = 0x - 14 = -14$$ 12. **Step 10:** Since the remainder $-14$ is degree 0, less than degree 1 of divisor, stop. 13. **Final answer:** $$\frac{4x^3 + x + 1}{2x + 3} = 2x^2 - 3x + 5 - \frac{14}{2x + 3}$$