1. **State the problem:** Find the first-order derivative of the function $$f(x) = (x^2 + 3)(x^2 - 2)$$. 2. **Recall the product rule:** For two functions $$u(x)$$ and $$v(x)$$, the derivative of their product is $$f'(x) = u'(x)v(x) + u(x)v'(x)$$. 3. **Identify functions:** Let $$u(x) = x^2 + 3$$ and $$v(x) = x^2 - 2$$. 4. **Compute derivatives:** - $$u'(x) = 2x$$ - $$v'(x) = 2x$$ 5. **Apply product rule:** $$f'(x) = u'(x)v(x) + u(x)v'(x) = 2x(x^2 - 2) + (x^2 + 3)2x$$ 6. **Simplify:** $$f'(x) = 2x(x^2 - 2) + 2x(x^2 + 3) = 2x(x^2 - 2 + x^2 + 3) = 2x(2x^2 + 1)$$ 7. **Final answer:** $$f'(x) = 2x(x^2 - 2) + 2x(x^2 + 3)$$ which matches the second option.