General antiderivative of a factored quadratic

Expand the product: $(x-2)(x+3) = x^2 + 3x - 2x - 6 = x^2 + x - 6$. This gives $f(x)$ in standard polynomial form, ready for term-by-term antidifferentiation.

Apply the power rule to each term: $\int x^2\,dx = \frac{x^3}{3}$, $\int x\,dx = \frac{x^2}{2}$, and $\int -6\,dx = -6x$. Adding a single arbitrary constant $C$ for the whole antiderivative gives $F(x) = \frac{x^3}{3} + \frac{x^2}{2} - 6x + C$.