So (x+y) (x+y+1/x+1/y) =10 (x+y).
(x+y)? +(x+y)/x +(x+y)/y= 10(x+y)
(x+y)? +y/x+x/y+2 = 10(x+y)( 1)
Because y/x +x/y≥2√[(y/x)(x/y)]=2, (take an equal sign if and only if y=x).
So (1) is expressed as
(x+y)? +2+2≤ 10(x+y)
That is (x+y)? - 10(x+y)+4≤0
The solution is 5-√2 1≤x+y≤5+√2 1,
So when x=y=(5+√2 1)/2, the maximum value of x+y is 5+√2 1.