mx²-2(m-2)x+(m+3)=0
m(x²-2x+1)=-(4x+3)
m(x-1)²=-(4x+3)
m=1时,左=0,右=-7,左≠右,等式不成立,因此x≠1
m=-(4x+3)/(x-1)²
m为非负整数,m+1为正整数,1-(4x+3)/(x-1)²为正整数
[(x-1)²-(4x+3)]/(x-1)²=(x²-6x-2)/(x²-2x+1)=k (k为正整数)
(k-1)x²-2(k-3)x+k+2=0
k=1时,等式变为4x+3=0 x=-3/4,方程有解
k>1时,方程有解,判别式Δ≥0
[-2(k-3)]²-4(k-1)(k+2)≥0
7k≤11
k≤11/7,k为正整数,只有k=1,此时只有x=-3/4
x=-3/4代入m=-(4x+3)/(x-1)²
m=-[4×(-3/4)+3]/(-3/4 -1)²=0
方程变为4x+3=0
x=-3/4
综上,得：非负整数m的值为0,方程所有的整数解只有一个：x=-3/4