Aparallelogram has coordinates a(1, 1), b(5, 4), c(7, 1), and d(3, -2). what are the coordinates of parallelogram a′b′c′d′ after a 180° rotation about the origin and a translation 5 units to the right and 1 unit down?

Take A  which is the point (1,1)
180 degree rotation takes it to (-1,-1)
Translation takes that to (-1+5, -1)  = (4,-1)
Then translation of 1 unit down   gives  the  final point at (4, -1-1) = (4,-2) = A'

If you pefrorm the same process for points B, C and  D you'll get the final answer.

The 180 rotation for B , C and D will take them to (-5,-4) , (-7,-1 ) and (-3, 2) 
respectively.