a)
(1 + 3x)^2 - (1 - 3x)^2=
=1+6x+9x^2-(1-6x+9x^2)=
=1+6x+9x^2-1+6x-9x^2=12x
b)
(1 - 2x^2)^2 - (2 - x^2)^2=
=1-4x^2+4x^4-(4-4x^2+x^4)=
=4x^4-4x^2+1-4+4x^2-x^4=
=3x^4-3
c)
(\sqrt5 - 3x^2) (\sqrt5 + 3x^2) + (1 + 3x^2)^2=
=(\sqrt5)^2-(3x^2)^2+1+6x^2+9x^4=
=5-9x^4+1+6x^2+9x^4=
=6x^2+6
d)
(\frac{\sqrt2}{2}x^2 + 1) (\frac{\sqrt2}{2} x^2 - 1) - \frac{1}{2}(x^2 + x)^2=
=(\frac{\sqrt2}{2}x^2)^2-1^2-\frac{1}{2}(x^4+2x^3+x^2)=
=\frac{2}{4}x^4-1-\frac{1}{2}x^4-x^3-\frac{1}{2}x^2=
=\frac{1}{2}x^4-1-\frac{1}{2}x^4-x^3-\frac{1}{2}x^2=
=-x^3-\frac{1}{2}x^2-1