I presume $P_2$ stands for the set of all polynomials of degree less than 2 over some already specified field, such as the real numbers for example.
They are asking you to determine whether the subsets of $P_2$ you have listed above satisfy the criterion for being a vector subspace (essentially, a vector space within a vector space). To do this you need to know what properties a vector space must satisfy and check whether these subsets satisfy these properties. Here goes:
A vector space is a set $V$, whose elements are called vectors, on which we have two operations, namely the binary operation of vector addition and scalar multiplication in the field $F$, satisfying the following axioms:
$(V,+)$ Axioms:
(1) $\vec{v}+\vec{w}=\vec{w}+\vec{v}$ for all $\vec{v},\vec{w} \in V$.
(2) $(\vec{u}+\vec{v})+\vec{w}=\vec{u}+(\vec{v}+\vec{w})$ for all $\vec{u},\vec{v},\vec{w} \in V$.
(3) There exists a vector $\vec{0} \in V$ such that $\vec{v}+\vec{0}=\vec{v}$ for all $\vec{v} \in V$.
(4) For every $\vec{v} \in V$ there exists a vector $-\vec{v}$ such that $\vec{v}+(-\vec{v})=\vec{0}$.
Scalar Multiplication Axioms:
(5) For all $\vec{v} \in V$, we have $1\cdot \vec{v}=\vec{v}$.
(6) $(ab)\vec{v}=a(b\vec{v})$ for all $a,b \in F$ and $\vec{v} \in V$.
Distributive Axioms:
(7) $(a+b)\vec{v}=a\vec{v}+b\vec{v}$ for all $a,b \in F$ and $\vec{v} \in V$.
(8) $a(\vec{v}+\vec{w})=a\vec{v}+a\vec{w}$ for all $a\in F$ and $\vec{v},\vec{w} \in V$.
Written using group theory terminology, a vector space is an abelian group $V$ acted upon by elements of a field. The nature of the actions are expressed in the scalar multiplication and distributive axioms.
Verifying that a subset is a subspace is not as tedious as checking all of the axioms above, because your subset automatically inherits all of the properties from the larger set, except for closure under the vector space operations. This is what you must check.