Answer
$W$ is not a vector subspace of $R^3$.
Work Step by Step
Let $W=\left\{\left(x_{1}, x_{2}, 4\right) : x_{1} \text { and } x_{2} \text { are real numbers }\right\}$, $u=(1,2,4)\in W$ and $c=2\in R$, then
\begin{align*}
cu&=2(1,2,4)\\
&=(2,4,8)\not\in W.
\end{align*}
Hence, $W$ is not a vector subspace of $R^3$.