\(A=(a+b+c)^3+(a-b-c)^3-6a(b+c)^2 \\=(a+b+c+a-b-c)[(a+b+c)^2-(a+b+c)(a-b-c)+(a-b-c)^2]-6a(b+c)^2 \\=2a[a^2+b^2+c^2+2ab+2ac+2bc-a^2+(b+c)^2+a^2+b^2+c^2-2ab-2ac+2bc-3(b+c)^2] \\=2a(a^2+2b^2+2c^2+4bc-2(b+c)^2] \\=2a(a^2+2b^2+2c^2+4bc-2b^2-4bc-2c^2) \\=2a.a^2=2a^3\)