Your set of polynomials (vectors) span p(x) if p(x) can be written as a linear combination of the vectors , i.e.

So you have to determine whether or not the system of equations (constructed by equating the coefficients of each term) is consistent or not:

Note that you don't actually have to find the solutions. The determinant will determine whether or not your system is consistent (implying your set of vectors span p(x)).