Writing the elements of this set as coordinate vectors, and using those vectors as columns of a matrix gives us:
$$\left(\begin{matrix}-3 & 2 & 1 \\ 0 & -1 & -2 \\ 1 & 0 & 1 \end{matrix}\right).$$
I contend that this matrix row reduces to the identity matrix, thus it is non-singular, thus the columns are linearly independent, thus the elements of the set given are linearly independent.
Proof that this set spans $P_2$ is left as an exercise for the reader.
Comments
Writing the elements of this set as coordinate vectors, and using those vectors as columns of a matrix gives us:
$$\left(\begin{matrix}-3 & 2 & 1 \\ 0 & -1 & -2 \\ 1 & 0 & 1 \end{matrix}\right).$$
I contend that this matrix row reduces to the identity matrix, thus it is non-singular, thus the columns are linearly independent, thus the elements of the set given are linearly independent.
Proof that this set spans $P_2$ is left as an exercise for the reader.