2x^3 + 250 =
The first rule of factoring is to always try to factor a common factor from all terms. The terms here are 2x^2 and 250. They have the factor 2 in common.
= 2(x^3 + 125)
Now you have a two-term polynomial.
The known two term polynomials that can be factored are a difference of two squares and a sum and a difference of two cubes. You obviously have a sum. x^2 and 125 are perfect cubes.
= 2(x^2 + 5^3)
You have a sum of cubes.
Now you must follow the pattern of the factorization of a sum of cubes which is:
a^3 + b^3 = (a + b)(a^2 - ab + b^2)
If you follow that pattern, you will get
= (x + 5)(x^2 - 5x + 25)