C1 Balculs \[ \begin{array}{l} \text { Sot } P=a+b+c+d, \quad Q=a+b-c-d, R=a-b+c-d \\ \text { at } S=a-d-c+d \\ \text { Balculer } \\ T=P Q\left(P^{2}+Q^{2}\right)-R S\left(R^{2}+S^{2}\right) \end{array} \]
Solution
Step 1 ：Given the expressions for P, Q, R, and S as follows: \(P = a + b + c + d\), \(Q = a + b - c - d\), \(R = a - b + c - d\), and \(S = a - c\)
Step 2 ：Substitute these expressions into the equation for T: \(T = PQ(P^2 + Q^2) - RS(R^2 + S^2)\)
Step 3 ：Simplify the equation to get: \(T = -(a - c)((a - c)^2 + (a - b + c - d)^2)(a - b + c - d) + ((a + b - c - d)^2 + (a + b + c + d)^2)(a + b - c - d)(a + b + c + d)\)
Step 4 ：Further simplify the equation by expanding the terms and combining like terms to get the final expression for T: \(T = 12a^3b + 4a^3d + 9a^2b^2 - 6a^2bd - 3a^2d^2 + 9ab^3 + 3ab^2d + 3abd^2 + ad^3 + 2b^4 - b^3c + 3b^2c^2 - 3b^2cd - 4bc^3 + 6bc^2d - 3bcd^2 - 12c^3d - 9c^2d^2 - 9cd^3 - 2d^4\)
Step 5 ：Final Answer: \(\boxed{T = 12a^3b + 4a^3d + 9a^2b^2 - 6a^2bd - 3a^2d^2 + 9ab^3 + 3ab^2d + 3abd^2 + ad^3 + 2b^4 - b^3c + 3b^2c^2 - 3b^2cd - 4bc^3 + 6bc^2d - 3bcd^2 - 12c^3d - 9c^2d^2 - 9cd^3 - 2d^4}\)
