(x^2/y+y^2/x)/(x/y+1)(y/x+1)=[(x^3+y^3/xy)/xy]/(x+y/y)(x+y/x)
=(x^3+y^3)/(x+y)^2
=(x+y)(x^2-xy+y^2)/(x+y)^2
=(x^2-xy+y^2)/(x+y)
Adding and subtracting 3xy in numerator, we get,
=(x^2-xy+y^2+3xy-3xy)/(x+y)
=[(x^2+2xy+y^2)-3xy]/(x+y)
=[(x+y)^2-3xy]/(x+y)
Substituting the values of x+y and xy, we get
=(64-33)/8
=31/8
Hence the result is proved