The vertices of the quadrilateral are A(-1,5), B(7,1), C(5, -3) and D(-3, 1). 

For the quadrilateral to be a rectangle the opposite sides should be parallel and the adjacent sides should be perpendicular.

The slope of the sides are:

AB: `(5 - 1)/(-1-7) = -1/2`

BC: `(1 +3)/(7-5) = 2`

CD: `(1+3)/(-3-5) = -1/2`

DA: `(5-1)/(-3+1) = 2`

Parallel lines have the same slope. This makes AB and CD, and BC and DA parallel. The product of the slope of perpendicular lines is equal to -1. As a result AB and BC, and CD and DA are perpendicular sides.

From the properties of the sides of the quadrilateral ABCD it is proved that it is a rectangle.