數學挑戰題ii

Let x1 =a x2=b x3=c x4=d :
(a+c)(a+d) = (b+c)(b+d) = n - 10
a , b are the roots of the equation :
(x+c)(x+d) = n - 10
x^2 + (c+d)x + (cd - n + 10) = 0
Sum of roots = a + b = - c - d
product of roots = ab = cd - n + 10
P = (a+c)(b+c) + (a+d)(b+d)
= ab + (a+b)c + c^2 + ab + (a+b)d + d^2
= 2ab + (a+b)(c+d) + c^2 + d^2
= 2cd - 2n + 20 - (c+d)^2 + c^2 + d^2
= 20 - 2n
= 2(10 - n)