ANSWER
Option C
[tex]\frac{2x+3}{x+2}[/tex]EXPLANATION
To find the division between two functions, first, we have to write the expression,
[tex]f(x)\div g(x)=\frac{2x^2-x-6}{x^2-4}[/tex]Then, to simplify, we have to factor each function by finding its zeros.
Function g(x) is a difference between two squares, so it can be factored as,
[tex]g(x)=(x+2)(x-2)[/tex]To find the zeros of function f(x) we can use the quadratic formula,
[tex]\begin{gathered} ax^2+bx+c=0 \\ \\ x=\frac{-b\pm\sqrt{b^2-4ac}}{2a} \end{gathered}[/tex]In this case, a = 2, b = -1, and c = -6,
[tex]x=\frac{-(-1)\pm\sqrt{(-1)^2-4\cdot2\cdot(-6)}}{2\cdot2}=\frac{1\pm\sqrt{1+48}}{4}=\frac{1\pm\sqrt{49}}{4}=\frac{1\pm7}{4}[/tex]So, the zeros of f(x) are,
[tex]\begin{gathered} x=\frac{1+7}{4}=\frac{8}{4}=2 \\ \\ x=\frac{1-7}{4}=\frac{-6}{4}=-\frac{3}{2} \end{gathered}[/tex]So, the factored form of f(x) is,
[tex]f(x)=2(x-2)\left(x+\frac{3}{2}\right)[/tex]Replace each function by its factored form in the quotient,
[tex]f(x)\div g(x)=\frac{2(x-2)(x+\frac{3}{2})}{(x+2)(x-2)}[/tex]The factor (x - 2) is common in both numerator and denominator, so it cancels out,
[tex]f(x)\div g(x)=\frac{2(x+\frac{3}{2})}{x+2}=\frac{2x+3}{x+2}[/tex]Hence, the quotient is,
[tex]f(x)\div g(x)=\frac{2x+3}{x+2}[/tex]