I have a cubic equation: $$-x^3 + 7x^2 - 16x + 12 = 0.$$
How they showed us to solve this quickly is to simplify the equation to $$-(x - 2)^2 (x - 3) = 0$$ and find the solutions this way.
My question is how to get to that form in a generic way.
I looked over at certain methods to get the solutions but they all seem to be taking too long and complicated.
In classes, cubic equations always have rational roots. This is because Cardano's method is such a mess. In your example, all the rational roots are among $\pm(1,2,3,4,6,12)$ so try them. If you find one, you have a quadratic-no problem. Outside class, a numeric solution will be acceptable and root finding will solve the problem, but you should first check the rational roots. If there is one, it may show you something about the problem.