a)
\frac{6x^4 + 5x^2}{-7x^3 + 2x^2}
-7x^3 + 2x^2\ne 0
-x^2(7x-2)\ne0
x\ne 0 , x\ne \frac{2}{7}
D \in R \ {0; 2/7}
\frac{6x^4 + 5x^2}{-7x^3 + 2x^2}=\frac{x^2(6x^2+5)}{x^2(-7x+2)}=\frac{6x^2+5}{-7x+2}=\frac{6x^2+5}{2-7x}
b)
\frac{15x^7-20x^5}{10x^5+30x^4}
10x^5+30x^4\ne 0
10x^4(x+3)\ne0
x\ne0 , x\ne -3
\frac{15x^7-20x^5}{10x^5+30x^4}=\frac{5x^5(3x^2-4)}{5x^4(2x+6)}=\frac{x(3x^2-4)}{2x-6}
c)
\frac{3x^3-x}{6x^4-2x^2}
6x^4-2x^2\ne 0
2x^2(3x^2-1)\ne 0
x\ne 0 i
3x^2= 1 => x^2=\frac{1}{3}
x\ne \frac{1}{\sqrt3} => x\ne \frac{\sqrt3}{3}
x\ne -\frac{1}{\sqrt3} => x\ne -\frac{\sqrt3}{3}
D \in R \ {0, \frac{\sqrt3}{3} , \frac{\sqrt3}{3}}
\frac{3x^3-x}{6x^4-2x^2}=\frac{x(3x^2-1)}{2x^2(3x^2-1)}=\frac{1}{2x}
d)
\frac{2x^6-5x^5}{15x^4-6x^3}
15x^4-6x^3\ne 0
3x^3(5x-2)\ne0
x\ne 0 , 5x\ne2 => x\ne \frac{2}{5}
\frac{2x^6-5x^5}{15x^4-6x^3}=\frac{x^5(2x-5)}{3x^3(5x-2)}=\frac{x^2(2x-5)}{3(5x-2)}