求证:当k不等于-1时,方程x^2+y^2+2kx+(4k+10)y+10k+20=0

配方,(x+k)^2+(y+2k+5)^2-5(k+1)^2=0.
当k不等于-1时,5(k+1)^2>0,圆心(-k,-2k-5),半径5^(1/2)*|k+1|.
对任意两个不同的k值,取m不相等n,且都不等于-1.
圆M(-m,-2m-5),半径r1=5^(1/2)*|m+1|;
圆N(-n,-2n-5),半径r2=5^(1/2)*|n+1|.
要证两圆相切,则证明 |MN|=r1+r2 即可.
验证,|MN|=[(m-n)^2+(2m-2n)^2]^(1/2)=5^(1/2)*|m-n|=r1+r2,得证.