1. **State the problem:** We are asked to perform polynomial division for the first problem: divide $$x^3 - 3x + 11$$ by $$x + 4$$. 2. **Recall the formula:** 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 that result, subtract, and repeat. 3. **Perform the division:** - Divide the leading term: $$\frac{x^3}{x} = x^2$$. - Multiply divisor by $$x^2$$: $$x^2(x + 4) = x^3 + 4x^2$$. - Subtract: $$\left(x^3 - 3x + 11\right) - \left(x^3 + 4x^2\right) = \cancel{x^3} - \cancel{x^3} - 4x^2 - 3x + 11 = -4x^2 - 3x + 11$$. 4. **Next step:** - Divide leading term: $$\frac{-4x^2}{x} = -4x$$. - Multiply divisor by $$-4x$$: $$-4x(x + 4) = -4x^2 - 16x$$. - Subtract: $$\left(-4x^2 - 3x + 11\right) - \left(-4x^2 - 16x\right) = \cancel{-4x^2} - \cancel{-4x^2} + 13x + 11$$. 5. **Next step:** - Divide leading term: $$\frac{13x}{x} = 13$$. - Multiply divisor by $$13$$: $$13(x + 4) = 13x + 52$$. - Subtract: $$\left(13x + 11\right) - \left(13x + 52\right) = \cancel{13x} - \cancel{13x} - 41 = -41$$. 6. **Conclusion:** The quotient is $$x^2 - 4x + 13$$ and the remainder is $$-41$$. **Final answer:** $$\frac{x^3 - 3x + 11}{x + 4} = x^2 - 4x + 13 + \frac{-41}{x + 4}$$