Answer
See explanation
Work Step by Step
Row reduce the augmented matrix for
\[
x_{1}\left[\begin{array}{l}
1 \\
a
\end{array}\right]+x_{2}\left[\begin{array}{c}
a \\
a+2
\end{array}\right]=\left[\begin{array}{l}
0 \\
0
\end{array}\right]
\]
Solve
\[
\left.\left[\begin{array}{cc}
1 & a \\
a & a+2
\end{array}\right]^{0}\right] \sim\left[\begin{array}{ccc}
1 & a & 0 \\
0 & a+2-a^{2} & 0
\end{array}\right]=\left[\begin{array}{ccc}
1 & a & 0 \\
0 & (2-a)(1+a) & 0
\end{array}\right]
\]
Observation
The system has a non-trivial solution only when
\[
(2-a)(1+a)=0
\]
Conclusion
The vectors are linearly independent for all $a$ except $a=2$ and $a=-1$
