1. **State the problem:** Differentiate the function $$f(x) = (2x^2 + 3)((x^5 - x + 2)^3)$$ with respect to $$x$$. 2. **Identify the rule:** This is a product of two functions, so we use the product rule: $$\frac{d}{dx}[u(x)v(x)] = u'(x)v(x) + u(x)v'(x)$$ where $$u(x) = 2x^2 + 3$$ and $$v(x) = (x^5 - x + 2)^3$$. 3. **Differentiate $$u(x)$$:** $$u'(x) = \frac{d}{dx}(2x^2 + 3) = 4x$$. 4. **Differentiate $$v(x)$$:** Use the chain rule for $$v(x) = [g(x)]^3$$ where $$g(x) = x^5 - x + 2$$. $$v'(x) = 3[g(x)]^2 \cdot g'(x)$$. Calculate $$g'(x)$$: $$g'(x) = 5x^4 - 1$$. So, $$v'(x) = 3(x^5 - x + 2)^2 (5x^4 - 1)$$. 5. **Apply the product rule:** $$\frac{d}{dx}f(x) = u'(x)v(x) + u(x)v'(x) = 4x (x^5 - x + 2)^3 + (2x^2 + 3) \cdot 3 (x^5 - x + 2)^2 (5x^4 - 1)$$. 6. **Factor common terms:** Both terms have $$ (x^5 - x + 2)^2 $$, so factor it out: $$\frac{d}{dx}f(x) = (x^5 - x + 2)^2 \left[4x (x^5 - x + 2) + 3(2x^2 + 3)(5x^4 - 1)\right]$$. 7. **Final answer:** $$\boxed{\frac{d}{dx}f(x) = (x^5 - x + 2)^2 \left[4x (x^5 - x + 2) + 3(2x^2 + 3)(5x^4 - 1)\right]}$$