a) (x-2)(x+2)(x^2+4)-(x^2-3)(x^2+3)
= (x^2-4)(x^2+4)-x^2-3^2
= x^2-4^2-x^2-3^2
=-4^2-3^2=-25
b) (a+b+c)^2 +(b+c-a)^2+(c+a-b)^2+(a+b-c)^2
=a^2+b^2+c^2+2ab+2ac+2bc
+b^2+c^2+a^2+2bc-2ba-2ca
+c^2+a^2+b^2+2ca-2cb-2ab
+a^2+b^2+c^2+2ab-2ac-2bc
=4a^2+4b^2+4c^2
=4(a^2+b^2+c^2)
c)(x-2)^3-(x+1)(x^2-x+1)+6(x-1)^2
=x^3-6x^2+12x-8-(x+1)^3+6(x^2-2X+1)
= x^3-6x^2+12x-8-(x^3+3x^2+3x+1)+6x^2-12x+6
=x^3-6x^2+12x-8-x^3-3x^2-3x-1+6x^2-12x+6
=-3x^2-3x-3
d)(x+1)^4-6(x+1)^2-(x^2-2)(x^2+2)
=(x+1)^2+(x+1)^2-6(x+1)^2-(x^2-2^2)
=(x+1)^2((x+1)^2-6)-x^2+4
=x^2+2x+1(x^2+2x-5)-x^2+4
=x^4+2x^3-5x^2+2x^3+4x^2-10x+x^2+2x-5-x^2+4
=x^4+4x^3-x^2-8x-1