Factor completely: 5x 3 + 8x 2 – 5x + 8

Are you sure that isn't 5x^{3}+8x^{2}-5x-8?

As it is with that +8 at the end it is not nicely factorable.

But with a -8, it is. Let me know before I proceed.

if the equation is as galactus had written, then proceed as follows;
5x^3+8x^2-5x-8
now see carefully-
(5x^3+8x^2)-(5x+8)
x^2(5x+8)-1(5x+8)
(x^2-1)(5x+8)
so;(if the equation is equal to zero)
x= +1,-1,-8/5
Dude, got it or not?

Remember, you can factor it further because of the x^{2}-1

(x+1)(x-1)(5x+8)

Yes! By using the formula-> a^2-b^2=(a-b)(a+b)