We have : 

2a+b+c+da=a+2b+c+db=a+b+2c+dc=a+b+c+2dd2a+b+c+da=a+2b+c+db=a+b+2c+dc=a+b+c+2dd

⇒2a+b+c+da−1=a+2b+c+db−1=a+b+2c+dc−1=a+b+c+2dd−1⇒2a+b+c+da−1=a+2b+c+db−1=a+b+2c+dc−1=a+b+c+2dd−1

⇒a+b+c+da=a+b+c+db=a+b+c+dc=a+b+c+dd⇒a+b+c+da=a+b+c+db=a+b+c+dc=a+b+c+dd

+> If a + b + c + d ≠0≠0

=> a = b = c = d (Because the same of namerator)

=> M = 1 + 1 + 1 + 1 = 4

+> If a + b + c + d = 0

=> a + b = -(c + d)

      b + c = -(a + d)

      c + d = -(a + b)

      a + d = -(b + c)

Change this into M , we have :

M=a+bc+d+b+cd+a+c+da+b+a+db+cM=a+bc+d+b+cd+a+c+da+b+a+db+c

M=−(c+d)c+d+−(a+d)a+d+−(a+b)a+b+−(b+c)b+c=(−1)+(−1)+(−1)+(−1)=−4M=−(c+d)c+d+−(a+d)a+d+−(a+b)a+b+−(b+c)b+c=(−1)+(−1)+(−1)+(−1)=−4

So M = 4 and M = -4

We have : 

2a+b+c+da=a+2b+c+db=a+b+2c+dc=a+b+c+2dd2a+b+c+da=a+2b+c+db=a+b+2c+dc=a+b+c+2dd

⇒2a+b+c+da−1=a+2b+c+db−1=a+b+2c+dc−1=a+b+c+2dd−1⇒2a+b+c+da−1=a+2b+c+db−1=a+b+2c+dc−1=a+b+c+2dd−1

⇒a+b+c+da=a+b+c+db=a+b+c+dc=a+b+c+dd⇒a+b+c+da=a+b+c+db=a+b+c+dc=a+b+c+dd

+> If a + b + c + d ≠0≠0

=> a = b = c = d (Because the same of namerator)

=> M = 1 + 1 + 1 + 1 = 4

+> If a + b + c + d = 0

=> a + b = -(c + d)

      b + c = -(a + d)

      c + d = -(a + b)

      a + d = -(b + c)

Change this into M , we have :

M=a+bc+d+b+cd+a+c+da+b+a+db+cM=a+bc+d+b+cd+a+c+da+b+a+db+c

M=−(c+d)c+d+−(a+d)a+d+−(a+b)a+b+−(b+c)b+c=(−1)+(−1)+(−1)+(−1)=−4M=−(c+d)c+d+−(a+d)a+d+−(a+b)a+b+−(b+c)b+c=(−1)+(−1)+(−1)+(−1)=−4

So M = 4 and M = -4