How do I factor these equations?

Question

I'm having trouble factoring a couple of problems
x^3+8
6n^2-11n-2
x^2+2xy+2x+y^2+2y-8
4x^6-4x^2
4a^2+12ab+9b^2-25c^2

Thanks!

Answer 1

x^3+ 8 is the sum of two cubes (8 is 2^3).
(x+2)(x^2-2x+4).

The second one is (6n +1)(n-2)

The third one you should reorder the terms...
x^2+2xy+y^2+2x+2y-8. The first three terms factor into
(x+y)(x+y)+2x+2y-8. Then simplify more to get
(x+y)^2+2(x+y)-8. Pretend (x+y)=a. Then you would get
a^2+2a-8. That's pretty easy to factor. It's
(a+4)(a-2). So replace a with (x+y) and the answer is
(x+y+4)(x+y-2).

The fourth one is 4x^2(x^4-1)=4x^2(x^2+1)(x^2-1)

The last one you take the first three terms and factor to get
(2a+3b)(2a+3b) -25c^2 which equals
(2a+3b)^2-25c^2 but 25c^2 can be written as (5c)^2 so you have
(2a+3b)^2-(5c)^2 which is the difference of two squares. So,
(2a+3b-5c)(2a+3b+5c) is the answer.