In Example BSP4 we showed that \begin{equation*} B=\set{x-2,\,x^2-4x+4,\,x^3-6x^2+12x-8,\,x^4-8x^3+24x^2-32x+16} \end{equation*} is a basis for $W=\setparts{p(x)}{p\in P_4,\ p(2)=0}$. So $\dimension{W}=4$.
The set \begin{equation*} \set{3x^2-5x-2,\,2x^2-7x+6,\,x^3-2x^2+x-2} \end{equation*} is a subset of $W$ (check this) and it happens to be linearly independent (check this, too). However, by Theorem G it cannot span $W$.
The set \begin{equation*} \set{3x^2-5x-2,\,2x^2-7x+6,\,x^3-2x^2+x-2,\,-x^4+2x^3+5x^2-10x,\,x^4-16} \end{equation*} is another subset of $W$ (check this) and Theorem G tells us that it must be linearly dependent.
The set \begin{equation*} \set{x-2,\,x^2-2x,\,x^3-2x^2,\,x^4-2x^3} \end{equation*} is a third subset of $W$ (check this) and is linearly independent (check this). Since it has the right size to be a basis, and is linearly independent, Theorem G tells us that it also spans $W$, and therefore is a basis of $W$.