We know the definition of span:

If

is a set of vectors in a vector space

, then the subspace

of

consisting of all linear combinations of the vectors in

is called the space spanned by

, and we say that the vectors

span

. To indicate that

is the space spanned by the vectors in the set

, we write:

or

I've a math problem and asked to find out whether the polynomials span a space or not. Determine whether the following polynomials span

(the set of degree 2 polynomials).

The book says it doesn't span

How do they figure out that

subspace does not consist of all linear combination of polynomials

? How does one come to this conclusion?