I am given two sub-spaces, v1 and v2. They are in the vector space $\mathbb{R}[x]_{\deg < 4}$.
$v_1=\text{span} \left( {x}^{3}+4\,{x}^{2}-x-3,{x}^{3}+5\,{x}^{2}+5 ,3\,{x}^{3}+10\,{x}^{2}-5\,x+3\right) $ $v_2=\text{span} \left( {x}^{3}-5\,x,{x}^{2}+x,{x}^{3}+2\,{x}^{2}-3 \,x\right)$
I am told that one sub-space is included in the other, and I need to
a. determine which subspace is included in the other
b. find the base of the smaller subspace
c. complete the base from the part b of the question so that it is the base of the larger subspace.
So far I've understood that $v_2$ is part of $v_1$ because $v_1$ has scalars without x-dependence, and $v_2$ does not have any. So $v_1$ includes $v_2$. Next for b I rref-ed the matrix of $v_2$ and found that the 3 vectors are linearly independent, and since I am told they are span therefore I know it is the base. Found. For c I need to add something so that it is the base of $v_1$. This is where I'm not certain how to proceed.