1. **State the problem:** Perform the long division of the polynomial $$x^8 - 4x^7 + 6x^5 - 2x^4 + 3x^2 + 4$$ by the divisor $$x^3 + 2$$. 2. **Recall the long division formula:** When dividing polynomials, we write $$\text{Dividend} = (\text{Divisor}) \times (\text{Quotient}) + \text{Remainder}$$ where the degree of the remainder is less than the degree of the divisor. 3. **Start dividing:** - Divide the leading term of the dividend $$x^8$$ by the leading term of the divisor $$x^3$$ to get $$x^{8-3} = x^5$$. 4. **Multiply and subtract:** - Multiply the divisor by $$x^5$$: $$x^5(x^3 + 2) = x^8 + 2x^5$$. - Subtract this from the dividend: $$\begin{aligned} &(x^8 - 4x^7 + 6x^5 - 2x^4 + 3x^2 + 4) - (x^8 + 2x^5) \\ &= \cancel{x^8} - 4x^7 + (6x^5 - 2x^5) - 2x^4 + 3x^2 + 4 \\ &= -4x^7 + 4x^5 - 2x^4 + 3x^2 + 4 \end{aligned}$$ 5. **Repeat the process:** - Divide the new leading term $$-4x^7$$ by $$x^3$$ to get $$-4x^4$$. - Multiply divisor by $$-4x^4$$: $$-4x^4(x^3 + 2) = -4x^7 - 8x^4$$. - Subtract: $$\begin{aligned} &(-4x^7 + 4x^5 - 2x^4 + 3x^2 + 4) - (-4x^7 - 8x^4) \\ &= \cancel{-4x^7} + 4x^5 + (-2x^4 + 8x^4) + 3x^2 + 4 \\ &= 4x^5 + 6x^4 + 3x^2 + 4 \end{aligned}$$ 6. **Next step:** - Divide $$4x^5$$ by $$x^3$$ to get $$4x^2$$. - Multiply divisor by $$4x^2$$: $$4x^2(x^3 + 2) = 4x^5 + 8x^2$$. - Subtract: $$\begin{aligned} &(4x^5 + 6x^4 + 3x^2 + 4) - (4x^5 + 8x^2) \\ &= \cancel{4x^5} + 6x^4 + (3x^2 - 8x^2) + 4 \\ &= 6x^4 - 5x^2 + 4 \end{aligned}$$ 7. **Continue:** - Divide $$6x^4$$ by $$x^3$$ to get $$6x$$. - Multiply divisor by $$6x$$: $$6x(x^3 + 2) = 6x^4 + 12x$$. - Subtract: $$\begin{aligned} &(6x^4 - 5x^2 + 4) - (6x^4 + 12x) \\ &= \cancel{6x^4} - 5x^2 - 12x + 4 \end{aligned}$$ 8. **Next:** - Divide $$-5x^2$$ by $$x^3$$ is not possible since degree of numerator is less than denominator, so stop here. 9. **Final result:** - Quotient: $$x^5 - 4x^4 + 4x^2 + 6x$$ - Remainder: $$-5x^2 - 12x + 4$$ **Answer:** $$\frac{x^8 - 4x^7 + 6x^5 - 2x^4 + 3x^2 + 4}{x^3 + 2} = x^5 - 4x^4 + 4x^2 + 6x + \frac{-5x^2 - 12x + 4}{x^3 + 2}$$