1. **State the problem:** We need to find the quotient and remainder when dividing the polynomial $$f(x) = 4x^4 - 33x^2 - 9x + 20$$ by $$x - 3$$. 2. **Formula and rule:** Polynomial division can be done using synthetic division or long division. The divisor is linear of the form $$x - c$$, so we use synthetic division with $$c = 3$$. 3. **Set up synthetic division:** Write coefficients of $$f(x)$$ in descending order of powers. Note the missing $$x^3$$ term has coefficient 0. Coefficients: $$4, 0, -33, -9, 20$$ 4. **Perform synthetic division:** - Bring down 4. - Multiply 4 by 3: 12. - Add to next coefficient: 0 + 12 = 12. - Multiply 12 by 3: 36. - Add to next coefficient: -33 + 36 = 3. - Multiply 3 by 3: 9. - Add to next coefficient: -9 + 9 = 0. - Multiply 0 by 3: 0. - Add to next coefficient: 20 + 0 = 20. 5. **Interpret result:** The bottom row (except last number) gives coefficients of quotient polynomial: $$4x^3 + 12x^2 + 3x + 0$$ The last number is the remainder: $$20$$. 6. **Final answer:** $$\text{Quotient} = 4x^3 + 12x^2 + 3x$$ $$\text{Remainder} = 20$$