1. **State the problem:** We need to divide the polynomial $$x^5 + 4x^3 - 11$$ by $$x - 1$$ using synthetic division and find the quotient and remainder. 2. **Recall synthetic division:** Synthetic division is a shortcut method for dividing a polynomial by a linear divisor of the form $$x - c$$. We use the root $$c$$ of the divisor to perform the division. 3. **Set up synthetic division:** Since the divisor is $$x - 1$$, we use $$c = 1$$. 4. **Write coefficients of the dividend:** The polynomial is $$x^5 + 0x^4 + 4x^3 + 0x^2 + 0x - 11$$, so coefficients are $$[1, 0, 4, 0, 0, -11]$$. 5. **Perform synthetic division:** - Bring down the first coefficient: $$1$$. - Multiply by $$c=1$$ and add to next coefficient: $$0 + 1 = 1$$. - Multiply by $$1$$ and add: $$4 + 1 = 5$$. - Multiply by $$1$$ and add: $$0 + 5 = 5$$. - Multiply by $$1$$ and add: $$0 + 5 = 5$$. - Multiply by $$1$$ and add: $$-11 + 5 = -6$$. 6. **Write the quotient and remainder:** The quotient coefficients are $$[1, 1, 5, 5, 5]$$ corresponding to $$x^4 + x^3 + 5x^2 + 5x + 5$$ and the remainder is $$-6$$. 7. **Check the user's given quotient and remainder:** The user states quotient $$1x^2 - 4x + 13$$ and remainder $$-41$$, which does not match the synthetic division result. 8. **Conclusion:** The correct quotient is $$x^4 + x^3 + 5x^2 + 5x + 5$$ and remainder $$-6$$ using synthetic division for $$\frac{x^5 + 4x^3 - 11}{x - 1}$$. **Final answer:** $$\text{Quotient} = x^4 + x^3 + 5x^2 + 5x + 5$$ $$\text{Remainder} = -6$$