Evaluate the sum of the following Legendre symbols: $\sum\limits_{a\in\mathbb{F}_p \ \text{and} \ a \neq 0,1} \left(\frac{a-a^2}{p} \right)$ 
Evaluate the sum of the following Legendre symbols: 
   $$\sum\limits_{a\in\mathbb{F}_p \ \text{and} \  a \neq 0,1} \left(\frac{a-a^2}{p} \right).$$

I am stuck in this step:  $\sum\limits_{a\in\mathbb{F}_p \ \text{and} \  a \neq 0,1} \left(\frac{-1}{p} \right)*(\frac{a^2 -a}{p})$
could anyone help me please?

You can seperate the cases when $p\equiv 3\pmod{4}$ and $p\equiv 1\pmod{4}$. Note that $$\sum_{a\in \mathbf{F}_{p}} \left(\frac{a^{2}-a}{p}\right) = \sum_{a \in \mathbf{F}_{p}} \left(\frac{a}{p}\right)\cdot \left(\frac{1-a}{p}\right) = \sum_{a \in \mathbf{F}_{p}} \left(\frac{a^2}{p}\right)\left(\frac{a^{-1}-1}{p}\right)$$
Now think what happens when to $a^{-1}-1$ as $a$ varies over $\mathbf{F}_{p}$.
