Bethany, lauren, amanda, and david all meet at a family reunion. they are comparing their ages. they have found that lauren is 13 years older than bethany and david is 11 years older than amanda. they also found that the product of bethany's and lauren's ages is equal to twice amanda's age. also, if they subtract 20 years from both bethany's and lauren's age, the product is equal to david's age.

As stated in the question 
Bethany - x
Amanda - y
the relationships are 
Lauren is 13 years older than Bethany 
Laurens age = x + 13
David is 11 years older than Amanda 
Davids age = y + 11
product of Bethany's and Laurens ages equal to twice Amandas age 
x * (x+13) = 2y
x² + 13x = 2y
divide all by 2
x²/2 + 13/2x = y
if they subtract 20 from Bethany and Laurens, then product is equal to Davids age
(x-20)(x+13-20) = y +11
(x-20)(x-7) = y+11
x² -7x -20x+140 = y+11
x² -27x +129 = y

the 2 equations to determine the relationships are 
y = x²/2 + 13/2x 
y = x² -27x +129
the first option is correct