Legendre symbol identity: $\sum_{a=1}^{p-1}a \cdot (\frac{a}{p} )$ and $\sum_{a=1}^{p-1}2^a \cdot (\frac{a}{p} )$

Question

I am trying to solve the following problems ($p$ is an odd prime).

1. Find the sum $$\sum_{a=1}^{p-1}a \cdot \left (\frac{a}{p} \right),$$
2. Find the sum $$\sum_{a=1}^{p-1} 2^a \cdot \left (\frac{a}{p} \right).$$

Some thoughts :

1. I reduced the sum to $2S_P-\frac{p(p-1)}{2}$ where $S_p$ is the sum of the quadratic residues modulo $p$ but I don't know how to evaluate it .
2. Nothing so far but more generally what can we say about the polynomial :

$$f(x)=\sum_{a=1}^{p-1} x^a \cdot \left (\frac{a}{p} \right)$$

Is this polynomial interesting in any way ?

Thanks for all the help .