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.