Do all polynomials of degree 2 form a subspace of $F(-\infty,\infty)$

Question

Let $F(-\infty,\infty)$ be a vector space, V. Do polynomials of degree 2 form a subspace of this vector space?

Answer 1

The zero element here is certainly not any polynomial of degree $2$, so it is not a subspace.

Answer 2

If you instead asked: "do all polynomials with degree two or less form a vector space", then the answer would be yes.

They wouldn't form a (multiplicative) algebra though.