Question

(5x^(1/3)+1)(25x^(2/3)-5x^(1/3)+1)

Answer 1

You probably do not recognize that as a "sum of cubles" facoring result, if you do, you can jump directly to the final line, otherwise ....

So let y = 5x^(1/3) , then 25x^(2/3) can be written as
(5x^(1/3))^2 = y^2

so we have (y + 1)(y^2 - y + 1)
= y^3 + 1
= (5x^(1/3))^3 + 1
= 125x + 1