If a,b,c,d  are positive reals such that a+b+c+d=2  and M=(a+b)(c+d),  then

The given conditions are a+b+c+d=2 ……. ( 1 ) M=(a+b)(c+d) …….. ( 2 )Let a+b=  α,  c+d=  β  Here a,b,c,d  are positive realsso that α,β  are Positive reals such that α+β =2  (from Eq. 1) and αβ=M  (from Eq. 2)Quadratic equation is x2−(α+β)x+αβ=0 ⇒x2−2x+M=0 Hence α,β  are real roots of the Equation  x2−2x+M=0. Its discriminant =(−2)2  −4.1. M≥0⇒  M≤1. (using formula of discriminant Δ = b2−4ac) Also,  M>0,  therefore 0