$\\$

$\bullet$ Trường hợp : `a+b+c+d = 0`

`-> a+b=0 - (c+d), b+c = - (d+a), c+d=0 - (a+b), d+a=0 - (b+c)`

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

`->M= (- (c+d) )/(c+d) + (- (d+a) )/(d+a) + ( -(a+b) )/(a+b) + (- (b+c) )/(b+c)`

`-> M = -1 + (-1)+(-1) + (-1)`

`-> M = -4`

Vậy `M=-4` khi `a+b+c+d=0`

$\\$

$\bullet$ Trường hợp : `a+b+c+d \ne 0`

Áp dụng tính chất dãy tỉ số bằng nhau có :

`(2019a +b+c+d)/a = (a+2019b +c+d)/b=(a+b+2019c+d)/c =(a+b+c+2019d)/d =(2019 a+b+c+d+a+2019b+c+d + a+b+2019c+d + a+b+c+2019d)/(a+b+c+d) = ( (2019a +a+a+a) + (b+2019b+b+b)+(c+c+2019c+c)+(d+d+d+2019d) )/(a+b+c+d) = (2022a + 2022b  + 2022c + 2022d)/(a+b+c+d) = (2022 (a+b+c+d) )/(a+b+c+d) = 2022`

`-> (2019a +b+c+d)/a=2022 ->2019a +b+c+d=2022a ->b+c+d=3a ->a+b+c+d=4a`

và `(a+2019b +c+d)/b = 2022 -> a+2019b  +c+d=2022b -> a+c+d=3b ->a+b+c+d=4b`

và `(a+b+2019c+d)/c=2022 ->a+ b+2019c +d=2022c -> a+b+d=3c ->a+b+c+d=4c`

và `(a+b+c+2019d)/d=2022 ->a+b+c+2019d=2022d ->a+b+c=3d ->a+b+c+d=4d`

Từ đó : `->4a=4b=4c=4d ->a=b=c=d`

`M = (a+b)/(c+d) + (b+c)/(d+a) + (c+d)/(a+b)+(d+a)/(b+c)`

`-> M = (a+a)/(a+a)+(b+b)/(b+b)+(c+c)/(c+d) + (d+d)/(d+d)`

`-> M = 1+1+1+1`

`-> M=4`

Vậy `M=4` khi `a+b+c+d \ne 0`

= $\frac{2019a+b+c+d}{a}$ -2018= $\frac{a+2019b+c+d}{b}$-2018 = $\frac{a+b+2019c+d}{c}$-2018 = $\frac{a+b+c+2019d}{d}$-2018

= $\frac{a+b+c+d}{a}$ = $\frac{a+b+c+d}{b}$ = $\frac{a+b+c+d}{c}$ = $\frac{a+b+c+d}{d}$

Từ đây ta suy ra 2 trường hợp:
+ Trường hợp 1:
Nếu a + b + c + d $\notin$ 0 $\longrightarrow$ a = b = c = d
$\longrightarrow$ M = 1 + 1 + 1 + 1 = 1 . 4 = 4
+ Trường hợp 2:
Nếu a + b + c + d = 0 thì
a + b = - ( c + d ) ; b + c = - ( d + a )
c + d = - ( a + b ) ; d + a = - ( b + c )
Do đó: M = ( -1 ) + ( - 1 ) + ( - 1 ) + ( - 1) = -4