Question 281596
I'll knock myself out ;)

a) {{{x^3-4x}}}
b) {{{xy-x+8y-8}}}
c) {{{x^3+x^2-90x}}}

a) We first want to look for any greatest common factors, notice there is an x in each term which is our greatest common factor!

{{{x^3 - 4x = x(x^2 - 4)}}} Notice how an x comes out of both terms
{{{x(x^2 - 4) = x(x+2)(x-2)}}} this is binomial factoring, we can do this because there is a minus sign in between {{{x^2}}} and {{{4}}}. also, both {{{4}}} and {{{x^2}}} can be square rooted perfectly.

So the final answer is:

{{{highlight(x(x+2)(x-2))}}}

b) This example is a super use of greatest common factoring, sometimes noted as gcd:

{{{xy-x+8y-8}}} group the first two together and the last two:

{{{(xy - x) + (8y - 8)}}} then factor by greatest common factoring:
{{{x(y-1) + 8(y - 1)}}} we can see that there are two y-1, we can factor that out too:
{{{(y-1) (x + 8)}}} the easiest way to see this step is to take out the y-1 and whatever is left over is your second term.
{{{highlight((y-1) (x + 8))}}} is your final answer

c) {{{x^3+x^2-90x}}}

Once again start off with greatest common factoring, there is an x is each of those:

{{{x(x^2 + x - 90)}}}
Now we want to find two things that multiply to be negative 90 and add up to be positive 1. One number is going to have to be a negative because that is the only way to multiply two numbers to get -90. The answer for this one is -9 and +10 which reduces to our final factoring:

{{{highlight(x (x + 10) (x - 9))}}} final answer