Determining whether the polynomials span $P_2$ 
Determine whether the following polynomials span $\mathbb P_2$
$p_1 = 1+x+2x^2$
$p_2 = 3+x$
$p_3 = 5-x+4x^2$
$p_4 = -2- 2x +2x^2$

Attempt: 
I created the following augmented matrix: 
$$
    \begin{bmatrix}
    1 & 3 & 5 & -2 & a \\
    x & x & -x & -2x&bx\\
    2x^2 & 0 & 4x^2 & 2x^2 & cx^2\\
    \end{bmatrix}
$$
which reduces to: 
$$
    \begin{bmatrix}
    1 & 3 & 5 & -2 & a \\
    0 & 1 & 3 & 0 &(a-b)/2\\
    0 & 0 & 1 & 1/2 &  (c+a-3b)/2\\
    \end{bmatrix}
$$
Now, I believe that the system is consistent and has infinitely many solutions so the polynomials span $\mathbb{P_2}$. 
Is this approach correct? 

If $P_2$ is the vector space of real polynomials of order smaller or equal to 2, then you can think of the given polynomials as vectors in $\mathbb {R^3}$ as;
$p_1=(1,1,2)$, $p_2=(3,1,0)$, $p_3=(5,-1,4)$, $p_4=(-2,-2,2)$ (Why?). Then check the rank of the following matrix: \begin{pmatrix}
1 & 1 & 2\\
3 & 1 & 0\\
5 & -1 & 4\\
-2 & -2 & 2
\end{pmatrix}
If the rank is equal to 3, then the answer is affirmative. 
