We have two functions, f(x) and g(x) and we have to find (f/g)(x).
Let h(x) = (f/g)(x).
We can write it like this:
[tex]h(x)=\frac{f(x)}{g(x)}=\frac{10x^3-12x^2+6x-7}{-14x^2-1}=\frac{-10x^3+12x^2-6x+7}{14x^2+1}[/tex](f/g)(x) is defined for all x but g(x)=0.
We can find the values of x for which (f/g) is not defined as:
[tex]\begin{gathered} -14x^2-1=0 \\ -14x^2=1 \\ x^2=-\frac{1}{14} \\ x=\sqrt[]{-\frac{1}{14}}\longrightarrow\text{not a real number} \end{gathered}[/tex]So we can conclude that (f/g)(x) is defined for all real numbers.
Answer: (f/g)(x) = (-10x^3+12x^2-6x+7)/(14x^2+1)
