Is W a subspace of P x P? Let $W \subset P \times P$ be given below, (here $P = \mathbb{R}[x]$)
$W = \{ (f(x), g(x)) \in P \times P| f(2) = 0 \land g^{\prime}(0) = 0 \}$. 
Is W a subspace of $P \times P$? Prove or disprove. 
I'm so lost, please help. 

What do you need to show $W$ satisfies in order to check it is a subspace?
Let's first show that $W$ is closed under addition and I'll leave you to check the other condition(s).
Let $(f,g), (p,q) \in W$. $(f,g)+(p,q) = (f+p,g+q)$.
$$(f+p)(0) = f(0)+p(0)=0+0$$
$$(g+q)'(0) = g'(0)+q'(0) = 0+0$$
So $(f+p,g+q) \in W$.
If you show that $W$ contains the zero element of $P \times P$ and that it's also closed under multiplication by scalars from $\mathbb{R}$, you're done.
