m²-2m+10=6n-n²
m²-2m+n²-6n+10=0
(m²-2m+1)+(n²-6n+9)=0
(m-1)²+(n-3)²=0
m-1=0 ,n-3=0
m=1 , n=3

原式=[(m²-n²)/(m²+2mn+n²)+2/(mn)÷(1/m+1/n)²]÷[1/(m+n)]
={ (m+n)(m-n)/(m+n)²+2/(mn)÷[(m+n)/(mn)]² }×(m+n)
={ (m-n)/(m+n)+2/(mn)×[(mn)²/(m+n)²] }×(m+n)
=[ (m-n)/(m+n)+2mn/(m+n)² ]×(m+n)
=[(m-n)/(m+n)]×(m+n)+[2mn/(m+n)²]×(m+n)
=m-n+2mn/(m+n)
=1-3+2×1×3/(1+3)
=-2+3/2
=-1/2