Hi Kalid. If you have to guess and check, it would be easier to do it directly by trying 1^2+1, 2^2+2, and so on. But as I searched the web, I found that there is no need to guess and check, you always need to subtract the higher number (at one side) and add the lower number (at the other side) if the goal is a positive number, and the opposite if it is negative.
Also I found that both numbers must have a difference of 1. Now is the main question: If I know 2 factors of the goal number (6 in our case) with a difference of 1, I already have the answer: x= the lower of the 2 factors. And I don’t need, and it doesn’t help, to make (x+3)*(x-2). The problem would be if I don’t know such 2 factors (anytime the goal is not 2,6,12,20,etc.), and for that you didn’t give any solution. It’s just like saying “To find a square root, first find two equal factors, than multiply by 1, and divide by 1, and you have the answer”
But I found a method that does help. It’s called completing the square: sqrt(y+.25)-.5=x (for positive x) where y is the goal number.

I would add that example to the body of your post to clarify the notion of finding values that eliminate the error in the error tracking form of the equation. This also promotes the idea of minimizing errors and converging on a final solution. As far as “why we factor” it’s to make it make identification of the solution more efficient. I do not need to “factor” to identify actual solutions. That seems to conflate mechanisms with solutions.