[Math] How to prove these summations for the legendre symbol

Question

How can I prove for the Legendre symbol that:

$$\sum_{a=1}^{p-1}{\left(\frac{a(a+1)}{p}\right)}= -1 = \sum_{b=1}^{p-1}{\left(\frac{(1+b)}{p}\right)}$$

Answer 1

Hint: Try adding and one to each sum, and reindexing the sum so that it looks like an orthogonality relation for characters.