Explain why no list of four polynomials spans P4(F)\mathcal P_4(\mathbf F).

The list (1,x,…,x4)(1,x,\dots,x^4) of length five is linearly independent in P4(F)\mathcal P_4(\mathbf F), by 2.23

(length of spanning list)≥5\text{(length of spanning list)} \ge 5

Therefor no list of length four can be spanning.