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
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