1. **State the problem:** Find the derivative of the function $$f(x) = \sqrt[3]{(x+2)^2} (3x - 14)$$ with respect to $$x$$. 2. **Rewrite the function:** Note that $$\sqrt[3]{(x+2)^2} = (x+2)^{\frac{2}{3}}$$, so $$f(x) = (x+2)^{\frac{2}{3}} (3x - 14)$$. 3. **Use the product rule:** For $$f(x) = u(x)v(x)$$, $$f'(x) = u'(x)v(x) + u(x)v'(x)$$. Here, $$u(x) = (x+2)^{\frac{2}{3}}$$ and $$v(x) = 3x - 14$$. 4. **Find $$u'(x)$$:** $$u'(x) = \frac{2}{3} (x+2)^{\frac{2}{3} - 1} = \frac{2}{3} (x+2)^{-\frac{1}{3}} = \frac{2}{3} \frac{1}{\sqrt[3]{x+2}}$$. 5. **Find $$v'(x)$$:** $$v'(x) = 3$$. 6. **Apply product rule:** $$f'(x) = u'(x)v(x) + u(x)v'(x) = \frac{2}{3} \frac{1}{\sqrt[3]{x+2}} (3x - 14) + (x+2)^{\frac{2}{3}} (3)$$. 7. **Rewrite $$f'(x)$$ with common denominator:** Express $$ (x+2)^{\frac{2}{3}} = (x+2)^{1} (x+2)^{-\frac{1}{3}} = (x+2) \frac{1}{\sqrt[3]{x+2}}$$. So, $$f'(x) = \frac{2}{3} \frac{3x - 14}{\sqrt[3]{x+2}} + 3 (x+2) \frac{1}{\sqrt[3]{x+2}} = \frac{2}{3} \frac{3x - 14}{\sqrt[3]{x+2}} + \frac{3(x+2)}{\sqrt[3]{x+2}}$$. 8. **Combine terms:** $$f'(x) = \frac{2}{3} \frac{3x - 14}{\sqrt[3]{x+2}} + \frac{3(x+2)}{\sqrt[3]{x+2}} = \frac{2(3x - 14)}{3 \sqrt[3]{x+2}} + \frac{3(x+2)}{\sqrt[3]{x+2}}$$. 9. **Find common denominator and add:** Multiply numerator and denominator of second term by 3: $$= \frac{2(3x - 14)}{3 \sqrt[3]{x+2}} + \frac{3 \cdot 3 (x+2)}{3 \sqrt[3]{x+2}} = \frac{2(3x - 14) + 9(x+2)}{3 \sqrt[3]{x+2}}$$. 10. **Simplify numerator:** $$2(3x - 14) + 9(x+2) = 6x - 28 + 9x + 18 = 15x - 10$$. 11. **Final derivative:** $$f'(x) = \frac{15x - 10}{3 \sqrt[3]{x+2}} = \frac{5(3x - 2)}{3 \sqrt[3]{x+2}}$$. 12. **Match with options:** This matches option (C). **Answer:** (C) $$f'(x) = \frac{5(3x - 2)}{3 \sqrt[3]{x+2}}$$.