help me find a basic for the next space? (10 points for the best answer)?

You want a BASIS for W. That is a set of vectors (elements of W) so that every element of W can be expressed uniquely has a linear combination of the basis vectors. If the base field is C, the complex numbers, then W is 2 dimensional, there will be two basis vectors {v1,v2} and:

W = { a*v1 + b*v2 | a, b any two complex numbers }

If the base field is the real numbers, the W is four dimensional.

Why is W only TWO dimensional? Polynomials of degree less than or equal to 3 are linear combinations of 1, x, x^2, and x^3, and hence are 4 dimensional. However there are two additional conditions: f(0)=0 and f(1)=0. They each describe hypersurfaces (3 dimensional spaces) in that 4 dimensional space and their intersection is 2 dimensional.

To solve the problem, how can someone describe polynomials that satisfy f(1)=0 and f(0)=0? Remember the FACTOR THEOREM, along with the Fundamental Theorem of Algebra. Applying those two theorems will give you a polynomial expression with two unknown constants, say "a" and "b". You should be able to rearrange into the form: f(x)=a*p(x)+b*q(x). "p" and "q" are your basis. If you get f(x) = a*p(x) + ab*q(x), then replace "ab" with "b".

A posible basis of W is B={1,x,x^2,x^3}

W = <1, x, x^2, x^3>

where <> means "the subspace generated by"