Suppose $p_0,\dots,p_m \in \mathcal P(\mathbf F)$ are such that $p_j$ has degree $j$. Prove that $p_0,\dots,p_m$ is a basis of $\mathcal P_m(\mathbf F)$.

$p_0$ is a constant, it's nonzero since if it were zero it would have degree $-\infty$ (see 2.12).

Our strategy will be to write the standard basis in terms of $p_0,\dots,p_m$ then apply 2.42.

We can write $1 = p_0/p_0$, now consider how $p_1 = ax + b$ for some $a,b \in \mathbf F$ with $a\ne 0$. We can write $x = (p_1 - b)/a$. Continue like this to write $x^k$ as a linear combination of of $p_0,\dots,p_k$. Thus $\text{span}(p_0,\dots,p_m) = \mathcal P_m(\mathbf F)$ and so $p_0,\dots,p_m$ is a basis by 2.42.