Answer
$W$ is a vector subspace of $R^3$.
Work Step by Step
Let $W=\{(a, b, a+2 b) \cdot a \text { and } b \text { are real pumbers }\}$, $u=(x_1, x_2, x_1+2 x_2),v=(y_1, y_2, y_1+2 y_2)\in W$, $c\in R$. Now, $W$ contains the zero vector and
\begin{align*}
u+v&=(x_1, x_2, x_1+2 x_2)+(y_1, y_2, y_1+2 y_2)\\
&= (x_1+y_1,x_2+ y_2, x_1+y_1+2( x_2+y_2))\in W.\\
\end{align*}
Also,
\begin{align*}
cu&=c(x_1, x_2, x_1+2 x_2)\\
&=(cx_1, cx_2, c(x_1+2 x_2))\in W.\\
\end{align*}
Hence, $W$ is a vector subspace of $R^3$.