The map $$\alpha_0 + \alpha_1 x + \cdots + \alpha_n x^n \mapsto \alpha_0 + \alpha_1 (x + 1) + \cdots + \alpha_n (x + 1)^n$$ defines an isomorphism from $F_n[x] = \{P \in F[x] \mid \textrm{deg} P \leq n\}$ into itself, where $F$ is some field.
To see that the map is surjective, given some polynomial $\beta_0 + \beta_1 x + \cdots + \beta_n x^n$, we must determine coefficients $\alpha_0, \dots, \alpha_n$ such that $$\alpha_0 + \alpha_1 (x + 1) + \alpha_2 (x + 1)^2 + \cdots + \alpha_n (x + 1)^n = \beta_0 + \beta_1 x + \cdots + \beta_n x^n.$$
This is not very hard to do; by applying the binomial theorem to the left-hand side and equating the resulting coefficients, we get the recurrence $$\alpha_{n - r} = \beta_{n - r} - \sum_{k = n - r + 1}^n {k \choose n - r} \alpha_k, \qquad r = 0, 1, \dots, n.$$ Computing backwards from $r = n$ to $r = 0$, we can uniquely transform the $\beta_k$'s into the $\alpha_k$'s.
Is there a name for this transformation of $\{\beta_k\}_{k = 0}^n$ into $\{\alpha_k\}_0^n$?