I find that the quotient rule is easier to work with algebraically, since you already have the lcd(which is the first step of simplifying a derivative found by the using the product on quotient.) Also, if you are interested in zeros/critical points, the quotient rule saves you steps on your...
because the solutions form a 2-dim vector space over the field of complex numbers, so by properties of vector spaces all scalar multiples are elements of said solution space, which in this case would include linear combinations of complex conjugates