Sum of first $n$ integer values of a polynomial as a polynomial in $n$

Question

Let $p$ be a polynomial. Let

$$S_p(n)=\sum_{k=1}^n p(k)$$

be the sum of the polynomial's values at the first $n$ integers.

Splitting the sum along each term's monomials, pulling the common coefficients before the sums and applying Faulhaber's formula yields result that

1. $S_p$ is a polynomial in $n$ and that

2. $\deg(S_p) = \deg(p) + 1$

This seems rather simple, so my question is: Do we really need to use the relatively heavy machinery of Faulhaber (giving exact coefficients) here or is there a way to immediately see that these two points are true?