In Example VRP2 we needed to know that \begin{equation*} D=\set{ -2-x+3x^2,\, 1-2x^2,\, 5+4x+x^2 } \end{equation*} is a basis for $P_2$. With Theorem CLI and Theorem CSS this task is much easier.
First, choose a known basis for $P_2$, a basis that forms vector representations easily. We will choose \begin{equation*} B=\set{1,\,x,\,x^2} \end{equation*}
Now, form the subset of $\complex{3}$ that is the result of applying $\vectrepname{B}$ to each element of $D$, \begin{align*} F&=\set{\vectrep{B}{-2-x+3x^2},\,\vectrep{B}{1-2x^2},\,\vectrep{B}{5+4x+x^2}}\\ &= \set{ \colvector{-2\\-1\\3},\, \colvector{1\\0\\-2},\, \colvector{5\\4\\1} } \end{align*} and ask if $F$ is a linearly independent spanning set for $\complex{3}$. This is easily seen to be the case by forming a matrix $A$ whose columns are the vectors of $F$, row-reducing $A$ to the identity matrix $I_3$, and then using the nonsingularity of $A$ to assert that $F$ is a basis for $\complex{3}$ (Theorem CNMB). Now, since $F$ is a basis for $\complex{3}$, Theorem CLI and Theorem CSS tell us that $D$ is also a basis for $P_2$.