1. **State the problem:** Divide the polynomial $$P(x) = x^4 - x^3 + 4x + 2$$ by the divisor $$D(x) = x^2 + 3$$ using polynomial long division. 2. **Recall the formula:** Polynomial division expresses $$P(x)$$ as $$$P(x) = D(x) \cdot Q(x) + R(x)$$ where $$Q(x)$$ is the quotient and $$R(x)$$ is the remainder with degree less than the degree of $$D(x)$$. 3. **Set up the division:** Divide $$x^4 - x^3 + 0x^2 + 4x + 2$$ by $$x^2 + 0x + 3$$. 4. **First division step:** Divide the leading term $$x^4$$ by $$x^2$$ to get $$x^2$$. Multiply $$x^2$$ by $$x^2 + 3$$: $$x^2 \cdot (x^2 + 3) = x^4 + 3x^2$$. Subtract this from the original polynomial: $$\begin{aligned} &(x^4 - x^3 + 0x^2 + 4x + 2) - (x^4 + 3x^2) \\ &= \cancel{x^4} - x^3 + 0x^2 + 4x + 2 - \cancel{x^4} - 3x^2 \\ &= -x^3 - 3x^2 + 4x + 2 \end{aligned}$$ 5. **Second division step:** Divide the new leading term $$-x^3$$ by $$x^2$$ to get $$-x$$. Multiply $$-x$$ by $$x^2 + 3$$: $$-x \cdot (x^2 + 3) = -x^3 - 3x$$. Subtract this from the current remainder: $$\begin{aligned} &(-x^3 - 3x^2 + 4x + 2) - (-x^3 - 3x) \\ &= \cancel{-x^3} - 3x^2 + 4x + 2 - \cancel{-x^3} + 3x \\ &= -3x^2 + 7x + 2 \end{aligned}$$ 6. **Third division step:** Divide the leading term $$-3x^2$$ by $$x^2$$ to get $$-3$$. Multiply $$-3$$ by $$x^2 + 3$$: $$-3 \cdot (x^2 + 3) = -3x^2 - 9$$. Subtract this from the current remainder: $$\begin{aligned} &(-3x^2 + 7x + 2) - (-3x^2 - 9) \\ &= \cancel{-3x^2} + 7x + 2 - \cancel{-3x^2} + 9 \\ &= 7x + 11 \end{aligned}$$ 7. **Conclusion:** The quotient is $$Q(x) = x^2 - x - 3$$ and the remainder is $$R(x) = 7x + 11$$. 8. **Final expression:** $$$x^4 - x^3 + 4x + 2 = (x^2 + 3)(x^2 - x - 3) + (7x + 11)$$