a) \(\left(a+b+c\right)^3-\left(b+c-a\right)^3-\left(a+c-b\right)^3-\left(a+b-c\right)^3\)
\(=\left[\left(a+b\right)+c\right]^3-\left[\left(b+c\right)-a\right]^3-\left[\left(a+c\right)-b\right]^3-\left[\left(a+b\right)-c\right]^3\)
\(=\left[\left(a+b\right)^3+3.\left(a+b\right)^2.c+3.\left(a+b\right).c^2+c^3\right]-\left[\left(b+c\right)^3-3.\left(b+c\right)^2.a+3.\left(b+c\right).a^2-a^3\right]-\left[\left(a+c\right)^3-3.\left(a+c\right)^2.b+3.\left(a+c\right).b^2-b^3\right]-\left[\left(a+b\right)^3-3.\left(a+b\right)^2.c+3.\left(a+b\right).c^2-c^3\right]\)\(=\left[\left(a^3+3a^2b+3ab^2+b^3\right)+3\left(a^2+2ab+b^2\right).c+3c^2a+3c^2b+c^3\right]-\left[\left(b^3+3b^2c+3bc^2+c^3\right)-3.\left(b^2+2bc+c^2\right).a+3a^2b+3a^2c-a^3\right]-\left[\left(a^3+3a^2c+3ac^2+c^3\right)-3\left(a^2+2ab+b^2\right).c+3c^2a+3c^2b-c^3\right]\)\(=\left(a^3+3a^2b+3ab^2+b^3+3ca^2+6abc+3b^2c+3c^2a+3c^2b+c^3\right)-\left(b^3+3b^2c+3bc^2+c^3-3ab^2-6abc-3ac^2+3a^2b+3a^2c-a^3\right)-\left(a^3+3a^2c+3ac^2+c^3-3a^2c-6abc-3b^2c+3c^2a+3c^2b-c^3\right)\)\(=a^3+3a^2b+3ab^2+b^3+3ca^2+6abc+3b^2c+3c^2a+3c^2b+c^3-b^3-3b^2c-3bc^2-c^3+3ab^2+6abc+3ac^2-3a^2b-3a^2c+a^3-a^3-3a^2c-3ac^2-c^3+3a^2c+6abc+3b^2c-3c^2a-3c^2b+c^3\)\(=3ab^2+6abc+3ab^2+6abc+a^3+6abc+3b^2c-3c^2b\)
\(=6ab^2+18abc+a^3+3b^2c-3bc^2\)
P/s: Ko chắc! Đây là kết quả của hơn 50 phút !
