Answer
$m_1=3.94kg, m_2=0.06kg$
Work Step by Step
Assume the two objects as point masses, with $$m_1=m$$ $$m_2=4.00kg-m$$ The gravitational force between the two masses is $$F=G\frac{m_1m_2}{r^2}=G\frac{m(4.00-m)}{r^2}=(6.67x10^{-11}\frac{Nm^2}{kg^2})\frac{4.00m-m^2}{(0.25)^2}=2.5x10^{-10}N$$
Rearrange the quadratic equation to become
$$m^2-4.00m+0.234=0$$
Use quadratic formula to solve for $m$, resulting in two values, which are the masses of the objects
$$m_1=3.94kg, m_2=0.06kg$$