$\text{Ta có:}$ `(2a+b+c+d)/a=(a+2b+c+d)/b = (a+b+2c+d)/c = (a+b+c+2d)/d`
`⇒ (2a+b+c+d)/a - 1=(a+2b+c+d)/b - 1 = (a+b+2c+d)/c - 1= (a+b+c+2d)/d - 1`
`⇒ (a+b+c+d)/a = (a+b+c+d)/b = (a+b+c+d)/c = (a+b+c+d)/d (1)`
`TH1: a + b + c + d \ne 0`
$\text{Từ (1)}$ `⇒ a=b=c=d`
`M = (a+b)/(c+d) + (b+c)/(d+a) + (c+d)/(a+b) + (d+a)/(b+c)`
`= (2a)/(2a) + (2b)/(2b) + (2c)/(2c) + (2d)/(2d)`
`= 1 + 1 + 1 + 1 = 4`
`TH2: a + b + c + d = 0`
`⇒ a + b = -(c + d)`
`b + c = -(a + d)`
`c+d = -(a + b)`
`d + a = -(b + c)`
`M = (a+b)/(c+d) + (b+c)/(d+a) + (c+d)/(a+b) + (d+a)/(b+c)`
`=(-(c+d))/(c+d) + (-(a+d))/(d+a)+(-(a+b))/(a+b)+(-(b+c))/(b+c)`
` = -1 + (-1) + (-1) + (-1) = -4`
