Answer
The mass of the lighter object is 46 kg and the mass of the heavier object is 104 kg.
Work Step by Step
Let $M$ be the mass of one object. Then the mass of the other object is $150~kg-M$. We can write an expression for the gravitational attraction between the two objects.
$F = \frac{G~M~(150~kg-M)}{R^2}$
$FR^2 = GM(150~kg) - GM^2$
$GM^2-GM(150~kg) +FR^2 = 0$
$(6.67\times 10^{-11}~m^3/kg~s^2)~M^2-(6.67\times 10^{-11}~m^3/kg~s^2)(150~kg)~M +(8.00\times 10^{-6}~N)(0.20~m)^2 = 0$
$(6.67\times 10^{-11}~m^3/kg~s^2)~M^2-(1.00\times 10^{-8}~m^3/s^2)~M +(3.2\times 10^{-7}~kg~m^3/s^2) = 0$
We can use the quadratic formula to find $M$.
$M = \frac{-b\pm \sqrt{b^2-4ac}}{2a}$
$M = \frac{-(-1.00\times 10^{-8}~m^3/s^2)\pm \sqrt{(-1.00\times 10^{-8}~m^3/s^2)^2-(4)(6.67\times 10^{-11}~m^3/kg~s^2)(3.2\times 10^{-7}~kg~m^3/s^2)}}{(2)(6.67\times 10^{-11}~m^3/kg~s^2)}$
$M = 46~kg, 104~kg$
The mass of the lighter object is 46 kg and the mass of the heavier object is 104 kg.
