Answer:
[tex]\text{4}\quad -3x^{-1}+x^{-2}[/tex]
Step-by-step explanation:
The answer choices suggest there is a common factor of (x+2) that can be removed from numerator and denominator.
The denominator factors as ...
[tex]x^3+2x^2=x^2(x+2)[/tex]
The numerator obviously has no factors of x. We can try any of several means to determine if (x+2) is a factor. I tried synthetic division and found that the numerator can be written as ...
[tex]-3x^2-5x+2=(-3x+1)(x+2)[/tex]
Then the expression simplifies to ...
[tex]\dfrac{-3x^2-5x+2}{x^3+2x^2}=\dfrac{(-3x+1)(x+2)}{x^2(x+2)}\\\\=\dfrac{-3x+1}{x^2}=\dfrac{-3x}{x^2}+\dfrac{1}{x^2}=-3x^{-1}+x^{-2} \qquad\text{matches choice 4}[/tex]
