Adding 0, multiplying by 1 and more... Throughout my entire life studying mathematics, up until this point, one thing that I have noticed that is key to mathematical problem-solving is knowing when and how to manipulate certain terms.
One such key technique is adding zero to any equation, or multiplying by one. I've seen applications of this in algebra, calculus, number theory and, in many cases, even geometry (where drawing auxiliary lines, as I tend to do, is very similar to this approach). Let's take a look at an easy example:
Suppose we have the following cubic equation and we want to solve for every real and complex root of $x$:
$x^3+x^2+2x-4=0$
One way to solve it is to add $0$, by adding and subtracting $8$ from this:
$x^3+x^2+2x-4+8-8=0$
By rearranging the terms and doing some basic manipulations, we get:
$(x^3-8)+(x^2+2x+4)=0$
$(x-2)(x^2+2x+4)+(x^2+2x+4)=0$
$(x-1)(x^2+2x+4)=0$
This can easily be solved now.
As you can see, by merely adding $0$, we managed to factor that seemingly challenging cubic.
My question is, how do we "know" when and how to add zero, multiply by one, etc in these scenarios? Is it some sort of intuition that you need to build? Is there a specific method to do this? Please be sure to also include how your thought process works when doing something like this.
 A: I sometimes tell my students that math is like the Monkey's Paw:  You can have whatever you wish, but there is a price to pay.  (In the Monkey's Paw stories, you wish for a million dollars.  You get your wish, but it's your son's life insurance. Then you wish your son back, and he comes back, but he's a zombie.)
When you look at a problem, you might notice that "if only there were a $2$ there, this would be much easier."  Well, your wish is granted, just put a plus $2$ there.  But make sure you pay the horrible price of subtracting the $2$ also.
In your example, because of your experience, you happen to notice that $x^2+2x+4$ is a factor of $x^3-2^3$, and since there's an $x^3$ sitting there, you brain wishes there was a $-8$.  So your wish is granted.  You just have to pay for your wish with a $+8$.
Besides experience and intuition, I think there is a fundamental principle here:  If you don't like the way things look, change them so that you like them better.  You have to pay for those changes, but you can put the payment elsewhere in the problem and maybe things will work out.
