In algebra how are polynomials factored?

Well, I typed out a very long thorough well thought out answer to your other post, but then I saw this one, so you're getting that answer here because it took me a long time to type out, and then if you want the answer to x²-x-20 then you have to scroll down to the bottom.

It depends on the degree of the polynomial. Most commonly you're working with binomials, which are second degree polynomials, and you factor them using the quadratic equation. Generally you have ax²+bx+c=0 (if your equation isn't equal to zero, subtract everything on one side of the equation from both sides of the equation), so the quadratic equation is:
(-b±√(b²-4ac))/2a

Phonetically that's, "Negative B, plus or minus the square root of B squared, minus 4 A C, all over 2 A."

This will give you two answers, one when you add negative B to the square root of B squared minus four times A times C, and one when you subtract negative B from the square root of B squared minus four times A times C. The way this works is, well, I'll give you an example.

Let's say you have x²+5x+6=0, well when you plug that in, you get:
(-5±√(5²-4(1)(6)))/2(1)
Well doing the arithmetic we get:
(-5±√(25-24))/2
Continuing we get:
(-5±√(1))/2
(-5±1)/2
-4/2 and -6/2
Well the quadratic equation gives you the values of x, so now we know x can equal both -2 and -3, so we write out the final equation as:
(x+2)(x+3)=0

so (x+2)(x+3)=x²+5x+6

Sometimes you'll have something like x²+6x=-9
Well before you can do the quadratic equation, you need to make it equal 0, so we add 9 to both sides, so we get x²+6x+9=0, and when we do the quadratic equation, we end up getting (-6±√(36-36))/2 which is -6±0/2 which is just -3 and -3, so we have (x+3)(x+3)=0 but you can write that as (x+3)²=0

There are equations for trinomials and quadnomials (3rd and 4th degree equations), but there are no equations for 5th degree polynomials or above. I know it sounds bizarre, but there's no equation to factor a 5th degree polynomial or higher, and if you continue your education to Calculus level 2, you will learn why.

And on a final note, check to see if b² is greater than or less than 4 times a times c, because if it's less, than you can't do the quadratic equation, because you can't take a square root of a negative number (with the exception of using i, but that's something you'll learn about later).

x²-x-20
Well a shortcut to factoring a binomial is to know how a binomial works.
ax²+bx+c
(x+d)(x+e)
Well b is d+e, and c is d times e.
So we have 1x²+(-1)x+(-20)
Well the factors of 20 are 1*20, 2*10, 4*5, 5*4, 10*2, and 20*1.
Obviously one of the two are going to be negative, because a positive times a negative is a negative. Now we see which of our factors are 1 number apart.
That would be 4*5 (or 5*4, but that's the same thing).
Now when we add them, we get 9, but like I said, one of them is negative, so...
-4+5 is 1, and 4+(-5) is -1, so there we go, the values are 4 and -5, so the factors are (x+4)(x-5).

And just for posterity sake, I'll do the quadratic equation too...
(-(-1)±√((-1)²-4(1)(-20)))/2(1)
(1±√(1+80))/2
(1±√(81))/2
(1±9)/2
10/2 and -8/2
5 and -4

Well the quadratic equation gives you the value of x when the equation is 0, so that means (x-5)(x+4)=0 so when you plug in either 5 or -4 you get 0.