Okay, I think I get it now. :-)

So to use a new example:

If we have the subspace P(4)=0 in P_2(R), and there are the following polynomials:

In order to satisfy P(4)=0 we know that 1 root is (x-4) since x=4 and inserted it gives us 4-4=0. Thus if we look at the polynomials and factorize them with this root, we can sort out which polynomials that fit as a basis:

The first polynomial satisfies since: (x-4)(-0.5-0.5)=4-x

The second polynomial cannot be factorized with (x-4) so we'll sort that one out.

The third polynomial satisfies: (x-4)(1-3)=8-2x

The fourth polynomial satisfies: (x-4)(x-4)=16-8x+x^2

The fifth polynomial satisfies aswell: (x-4)(x-3)=12-7x+x^2

So 4 of our 5 polynomials seem to be a basis in our subspace which means that our subspace has 4 elements and thus 4 dimensions.

Was this a correct procedure?