a)
(2x^4-3x+5)(x^3+4x+1)=2x^7+8x^5+2x^4-3x^4-12x^2-3x+5x^3+20x+5=
2x^7+8x^5-x^4+5x^3-12x^2+17x+5
b)
(x+1)^2(x-2)=(x^2+2x+1)(x-2)=x^3-2x^2+2x^2-4x+x-2=x^3-3x-2
c)
\frac{4x^2-1}{x^2}*\frac{x}{2x^2+x}=\frac{(2x-1)(2x+1)}{x}*\frac{1}{2x^2+x}=\frac{(2x-1)(2x+1)}{2x^3+x^2}=
\frac{(2x-1)(2x+1)}{x^2(2x+1)}=\frac{2x-1}{x^2}
d)
\frac{2x-6}{x}:\frac{3x-9}{4x^2}=\frac{2(x-3)}{x}*\frac{4x^2}{3x-9}=2(x-3)*\frac{4x}{3(x-3)}=\frac{2*4x}{3}=\frac{8x}{3}
e)
\frac{x}{x-2}-\frac{x+2}{2x-4}=\frac{x}{x-2}-\frac{x+2}{2(x-2)}=\frac{2x-(x+2)}{2(x-2)}=\frac{2x-x-2}{2(x-2)}=\frac{x-2}{2(x-2)}=\frac{1}{2}