Don't Guess 1

Find all the positive integral solutions to the equation $y^2 = x^2 + x + 2$ .

If the solutions of $(x,y)$ are $(x_1, y_1) , (x_2, y_2) , \ldots, (x_n, y_n)$ , find $\displaystyle \sum_{i=1}^n x_i y_i$ .