Quadrilateral ABCD is a parallelogram because both pairs of opposite sides are parallel.
Slope of a line is defined as the difference in Y divided by the difference in X. It doesn't matter which way you do the subtraction as long as you do it in the same order for both X and Y. So let's calculate the slope of AB
A(-5,1)
B(-2,5)
(1 - 5)/(-5 - (-2)) = -4/-3 = 4/3
Slope BC
B(-2,5)
C(5,3)
(5-3)/(-2-5) = 2/-7 = -2/7
Slope CD
C(5,3)
D(2,-1)
(3-(-1))/(5-2) = 4/3
Slope AD
A(-5,1)
D(2,-1)
(1-(-1))/(-5-2) = 2/-7 = -2/7
So our slopes are 4/3, -2/7, 4/3, and -2/7
Since the slopes of AB and CD are identical, those lines are parallel. And since the slopes of both BC and AD are identical, those 2 lines are also parallel. But since 4/3 * -2/7 = -8/21 which is NOT -1, that means that lines AB and BC are NOT perpendicular, so you don't have a rectangle. But you do have a parallelogram because both pairs of opposite sides are parallel.
