Explain why there does not exist a list of six polynomials that are linearly independent in $\mathcal P_4(\mathbf F)$.

The list $(1,x,\dots,x^4)$ in $\mathcal P_4(\mathbf F)$ is spanning, by 2.23 the length of any linearly independent list $\le$ length of any spanning list. Meaning we can have at most five linearly independent vectors in $\mathcal P_4(\mathbf F)$, and so a list of six cannot exist.