1. **State the problem:** Divide the polynomial $4x^4 - 33x^2 - 9x + 2$ by $p(x) = x - 3$ using polynomial long division. 2. **Recall the division formula:** For polynomials, dividing $f(x)$ by $p(x)$ gives a quotient $q(x)$ and remainder $r(x)$ such that: $$f(x) = p(x) \cdot q(x) + r(x)$$ where the degree of $r(x)$ is less than the degree of $p(x)$. 3. **Set up the division:** Write $4x^4 - 33x^2 - 9x + 2$ in standard form including all powers: $$4x^4 + 0x^3 - 33x^2 - 9x + 2$$ 4. **Divide the leading term:** Divide $4x^4$ by $x$ to get $4x^3$. This is the first term of the quotient. 5. **Multiply and subtract:** Multiply $4x^3$ by $x - 3$: $$4x^3 \cdot (x - 3) = 4x^4 - 12x^3$$ Subtract this from the original polynomial: $$(4x^4 + 0x^3 - 33x^2 - 9x + 2) - (4x^4 - 12x^3) = 0x^4 + 12x^3 - 33x^2 - 9x + 2$$ 6. **Repeat the process:** Divide $12x^3$ by $x$ to get $12x^2$. Multiply $12x^2$ by $x - 3$: $$12x^2 \cdot (x - 3) = 12x^3 - 36x^2$$ Subtract: $$(12x^3 - 33x^2) - (12x^3 - 36x^2) = 0x^3 + 3x^2$$ Bring down the remaining terms: $$3x^2 - 9x + 2$$ 7. **Continue dividing:** Divide $3x^2$ by $x$ to get $3x$. Multiply $3x$ by $x - 3$: $$3x \cdot (x - 3) = 3x^2 - 9x$$ Subtract: $$(3x^2 - 9x) - (3x^2 - 9x) = 0$$ Bring down the $+2$: $$2$$ 8. **Final step:** Divide $2$ by $x$ is not possible (degree of remainder less than divisor), so remainder is $2$. 9. **Write the result:** $$\text{Quotient } q(x) = 4x^3 + 12x^2 + 3x$$ $$\text{Remainder } r(x) = 2$$ 10. **Check:** $$f(x) = (x - 3)(4x^3 + 12x^2 + 3x) + 2$$ **Final answer:** $$\boxed{\frac{4x^4 - 33x^2 - 9x + 2}{x - 3} = 4x^3 + 12x^2 + 3x + \frac{2}{x - 3}}$$