`text{Ta có :}`
$\dfrac{2a + b + c + d}{a}$ `=` $\dfrac{a + 2b + c + d}{b}$
`=` $\dfrac{a + b + 2c + d}{c}$ `=` $\dfrac{a + b + c + 2d}{d}$
`=>` $\dfrac{a + b + c + d}{a}$ `+ 1 =` $\dfrac{a + b + c + d}{b}$ `+ 1`
`=` $\dfrac{a + b + c + d}{c}$ `+ 1 =` $\dfrac{a + b + c + d}{d}$ `+ 1`
`=>` $\dfrac{a + b + c + d}{a}$ `=` $\dfrac{a + b + c + d}{b}$
`=` $\dfrac{a + b + c + d}{c}$ `=` $\dfrac{a + b + c + d}{d}$ `( 1 )`
`text{Nếu}` `a + b + c + d = 0`
=> $\left[\begin{matrix} a + b = -(c + d)\\ b + c = -(a + d)\\c + d = -(a + b)\\a + d = -(b + c )\end{matrix}\right.$
`text{Ta có}`
`M =` $\dfrac{a+b}{c+d}$ `+` $\dfrac{b+c}{d+a}$ `+` $\dfrac{c+d}{a+b}$ `+` $\dfrac{a+d}{b+c}$
`=> M = ` $\dfrac{-(c+d)}{c+d}$ `+` $\dfrac{-(a+d)}{d+a}$ `+` $\dfrac{-(a+b)}{a+b}$ `+` $\dfrac{-(b+c)}{b+c}$
`=> M = (-1) + (-1) + (-1) + (-1)`
`=> M = -4`
`text{Nếu}` `a + b + c + d` $\ne$ `0`
`=> a = b = c = d`
`=> M =` $\dfrac{a+b}{c+d}$ `+` $\dfrac{b+c}{d+a}$ `+` $\dfrac{c+d}{a+b}$ `+` $\dfrac{a+d}{b+c}$
`= 1 + 1 + 1 + 1`
`=> M = 4`
`text{Vậy}` `: M ∈ { 4 ; -4 }`
