1. **State the problem:** We need to find the derivative of the function $$F(x) = (6x^2 + 2x)(3x + 2)^2$$ with respect to $$x$$. 2. **Recall the product rule:** For two functions $$u(x)$$ and $$v(x)$$, the derivative of their product is given by: $$\frac{d}{dx}[u(x)v(x)] = u'(x)v(x) + u(x)v'(x)$$ 3. **Identify $$u$$ and $$v$$:** Let $$u = 6x^2 + 2x$$ and $$v = (3x + 2)^2$$. 4. **Find $$u'$$:** $$u' = \frac{d}{dx}(6x^2 + 2x) = 12x + 2$$. 5. **Find $$v'$$:** Use the chain rule for $$v = (3x + 2)^2$$: $$v' = 2(3x + 2) \cdot \frac{d}{dx}(3x + 2) = 2(3x + 2)(3) = 6(3x + 2)$$. 6. **Apply the product rule:** $$F'(x) = u'v + uv' = (12x + 2)(3x + 2)^2 + (6x^2 + 2x)6(3x + 2)$$. 7. **Simplify:** First term: $$(12x + 2)(3x + 2)^2$$ Second term: $$6(6x^2 + 2x)(3x + 2) = (36x^2 + 12x)(3x + 2)$$ 8. **Expand second term:** $$(36x^2)(3x) + (36x^2)(2) + (12x)(3x) + (12x)(2) = 108x^3 + 72x^2 + 36x^2 + 24x = 108x^3 + 108x^2 + 24x$$ 9. **Final derivative expression:** $$F'(x) = (12x + 2)(3x + 2)^2 + 108x^3 + 108x^2 + 24x$$ This is the derivative of the given function.