The functions we have are:
[tex]\begin{gathered} f(x)=2^{}x^2-x-6 \\ g(x)=x^2-4 \end{gathered}[/tex]
The division of the functions is:
[tex]\frac{f(x)}{g(x)}=\frac{2x^2-x-6}{x^2-4}[/tex]
We need to factor both the numerator f(x) and the denominator g(x) to find our answer.
Factoring 2x^2-x-6:
[tex]2x^2-x-6=(2x+3)(x-2)[/tex]
Factoring x^2-4:
[tex]x^2-4=(x+2)(x-2)[/tex]
And we substitute this in the division:
[tex]\frac{f(x)}{g(x)}=\frac{(2x+3)(x-2)}{(x+2)(x-2)}[/tex]
x-2 in the numerator and denominator cancel each other, and the final result is:
[tex]\frac{f(x)}{g(x)}=\frac{2x+3}{x+2}[/tex]
Option C
