\(\frac{2a+b+c+d}{a}=\frac{a+2b+c+d}{b}=\frac{a+b+2c+d}{c}=\frac{a+b+c+2d}{c}\)
\(\Rightarrow\frac{2a+b+c+d}{a}-1=\frac{a+2b+c+d}{b}-1=\frac{a+b+2c+d}{c}-1=\frac{a+b+c+2d}{d}-1\)
\(\Rightarrow\frac{a+b+c+d}{a}=\frac{a+b+c+d}{b}=\frac{a+b+c+d}{c}=\frac{a+b+c+d}{d}\)
\(=\frac{a+b+c+d+a+b+c+d+a+b+c+d+a+b+c+d}{a+b+c+d}=4\)
Xét \(a+b+c+d=0\)
\(\Rightarrow a+b=-\left(c+d\right),b+c=-\left(a+d\right),c+d=-\left(b+a\right),d+a=-\left(c+b\right)\)
\(\Rightarrow M=\frac{-\left(c+d\right)}{c+d}+\frac{-\left(d+a\right)}{d+a}+\frac{-\left(a+b\right)}{a+b}+\frac{-\left(b+c\right)}{b+c}\)
\(M=-1+-1+-1+-1=-4\)
Xét \(a+b+c+d\ne0\Rightarrow a=b=c=d\)
\(\Rightarrow M=1+1+1+1=4\)
Vậy M=-4 hoặc M=4
