Question

I need help with this problem.

Answer 1

Answer:

  $\text{4}\quad -3x^{-1}+x^{-2}$

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 ...

  $x^3+2x^2=x^2(x+2)$

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 ...

  $-3x^2-5x+2=(-3x+1)(x+2)$

Then the expression simplifies to ...

  $\dfrac{-3x^2-5x+2}{x^3+2x^2}=\dfrac{(-3x+1)(x+2)}{x^2(x+2)}\\\\=\dfrac{-3x+1}{x^2}=\dfrac{-3x}{x^2}+\dfrac{1}{x^2}=-3x^{-1}+x^{-2} \qquad\text{matches choice 4}$