, so a basis for would consist of 3 linearly independent vectors, so you would just have to remove one vector from your list and then show that the others are independent and span an arbitrary polynomial.
What you had:
If we want to see if the vectors span any arbitrary polynomal , then we have the matrix
What might be simpler:
Try constructing r as a linear combination of p and q, then remove from the list. then apply the same logic as above with the linear combination of p,q,s to see what the constants have to be (dependent on a,b,d in the polynomial)