1. **State the problem:** We are given the equation $$x^4 + x^3 + x^2 + x + 1 = 0$$ and asked to find the value of $$\left(x^{33} + \frac{2}{x^{22}}\right)\left(x^{22} + \frac{3}{x^{33}}\right).$$ 2. **Analyze the given polynomial:** The polynomial $$x^4 + x^3 + x^2 + x + 1 = 0$$ is a geometric series sum for $$x \neq 1$$: $$\frac{x^5 - 1}{x - 1} = 0 \implies x^5 = 1, \quad x \neq 1.$$ 3. **Important rule:** Since $$x^5 = 1$$ and $$x \neq 1$$, $$x$$ is a 5th root of unity other than 1. This implies: $$x^5 = 1, \quad x^0 = 1, \quad x^{5k} = 1 \text{ for any integer } k.$$ 4. **Simplify powers using $$x^5=1$$:** - $$x^{33} = x^{(5 \times 6) + 3} = x^3$$ - $$x^{22} = x^{(5 \times 4) + 2} = x^2$$ - $$\frac{1}{x^{22}} = x^{-22} = x^{-(5 \times 4 + 2)} = x^{-2} = x^{3}$$ (since $$x^{-2} = x^{5-2} = x^3$$) - $$\frac{1}{x^{33}} = x^{-33} = x^{-(5 \times 6 + 3)} = x^{-3} = x^{2}$$ (since $$x^{-3} = x^{5-3} = x^2$$) 5. **Rewrite the expression:** $$\left(x^{33} + \frac{2}{x^{22}}\right)\left(x^{22} + \frac{3}{x^{33}}\right) = (x^3 + 2x^3)(x^2 + 3x^2) = (3x^3)(4x^2) = 12x^{5}.$$ 6. **Use $$x^5=1$$:** $$12x^5 = 12 \times 1 = 12.$$ **Final answer:** $$\boxed{12}$$