I've attempted to solve the problem below, and here is what I got for a solution:
Given $f(x)=x^2-9$ and $g(x)=x^2+3x-1$, find $(fg)(x).$
$$ \begin{align} (fg)(x)&=(x^2-9)(x^2+3x-1)\\ &=x^4+3x^3-x^2-9x^2-27x+9\\ &=x^4+3x^3-10x^2-27x+9 \end{align} $$
Have I done this correctly?
I am wondering if I should have factored $(x^2-9)$ before multiplying, but I'm not sure if it would have made a difference.
You are correct. If I know what you are saying, you are right that it wouldn't matter. The factored form is $(x+3)(x-3)(x^2+3x-1)$.