The equation is

#m^2x^2+(2m-m^2)x+3=0#

This means that for the two roots #alpha# and #beta# we have

#alpha+beta = (m^2-2m)/m^2 qquad"and"qquad alpha beta = 3/m^2#

Thus

#alpha/beta + beta/alpha = (alpha^2+beta^2)/(alpha beta)#

#qquadqquad = ((alpha+beta)^2-2alpha beta)/(alpha beta)#

#qquad qquad = (alpha+beta)^2/(alpha beta)-2#

#qquadqquad = (m^2-2m)^2/m^4 times m^2/3-2#

#qquadqquad = (m^2-2m)^2/(3m^2)-2#

#qquad qquad = (m-2)^2/3-2#

(note that #m ne 0# as is obvious from the original equation, which leads to a contradiction for #m=0#)

and so

#alpha/beta + beta/alpha = 4/3 implies#

#(m-2)^2/3-2=4/3 implies#

#(m-2)^2 = 10 implies#

#m^2-4m-6=0#

(note that we can easily find the two values #m_1# and #m_2# to be #2 pm sqrt 10#, respectively - but the algebra is easier if we proceed by using properties of the quadratic equation)

From this quadratic we have

#m_1+m_2 = 4 qquad"and"qquad m_1m_2=-6#

Hence

#m_1^2/m_2++m_2^2/m_1 = (m_1^3+m_2^3)/(m_1m_2)#

#qquad qquad = ((m_1+m_2)^3-3m_1m_2(m_1+m_2))/(m_1m_2)#

#qquadqquad = (m_1+m_2)^3/(m_1m_2)-3(m_1+m_2)#

#qquadqquad = 4^3/(-6)-3times 4 = -64/6-12 = -68/3#