$P_4$ is the vector space of polynomials with degree at most $4$ (Example VSP). Define a subset $W$ as \begin{equation*} W=\setparts{p(x)}{p\in P_4,\ p(2)=0} \end{equation*} so $W$ is the collection of those polynomials (with degree 4 or less) whose graphs cross the $x$-axis at $x=2$. Whenever we encounter a new set it is a good idea to gain a better understanding of the set by finding a few elements in the set, and a few outside it. For example $x^2-x-2\in W$, while $x^4+x^3-7\not\in W$.
Is $W$ nonempty? Yes, $x-2\in W$.
Additive closure? Suppose $p\in W$ and $q\in W$. Is $p+q\in W$? $p$ and $q$ are not totally arbitrary, we know that $p(2)=0$ and $q(2)=0$. Then we can check $p+q$ for membership in $W$, \begin{align*} (p+q)(2)&=p(2)+q(2)&&\text{Addition in }P_4\\ &=0+0&&p\in W,\,q\in W\\ &=0 \end{align*} so we see that $p+q$ qualifies for membership in $W$.
Scalar multiplication closure? Suppose that $\alpha\in\complex{\null}$ and $p\in W$. Then we know that $p(2)=0$. Testing $\alpha p$ for membership, \begin{align*} (\alpha p)(2)&=\alpha p(2)&&\text{Scalar multiplication in }P_4\\ &=\alpha 0&&p\in W\\ &=0 \end{align*} so $\alpha p\in W$.
We have shown that $W$ meets the three conditions of Theorem TSS and so qualifies as a subspace of $P_4$. Notice that by Definition S we now know that $W$ is also a vector space. So all the properties of a vector space (Definition VS) and the theorems of Section VS apply in full.