How Do First Derivatives Help Identify Critical Points and Local Extrema?

[PowerBuilder]

I don't post many questions here since I'm usually able to find most of my answers before starting a thread. As only a few of you may know I'm gradually teaching myself calculus. What can I say? I guess I'm that kind of guy.

I've managed to pick up on quite a few good basic ideas. There is one question I have about 'first derivatives'...

The derivative of ( 4x^3 ) is 12x^2. I understand that. How in the heck though does the below example work, in regards to the introduction of the x^2 - x - 2 in brackets?

I'm using 'calculus for dummies' as one source of reading material, besides the entire internet...but can't seem to find out...whats happened. I've missed a beat somewhere.

The function f(x)=3x^4 − 4x^3−12x^2+3 has first derivative

f(x) = 12x^3−12x^2−24x
= 12x (x^2 − x −2)
= 12x (x + 1)(x− 2)

Source - Harvey Mudd College

 

[lavinia]

Just factor 12x out of each term

 

[S_Happens]

That part is just algebra. They are factoring it algebraically to find critical points (where the first derivative is zero or undefined). As that page shows, it's useful for finding local maximum and minimum points of the original function.