Answer:
The answer <em>should </em>be:
![=\frac{8x^3-18x}{3x+2}](https://tex.z-dn.net/?f=%3D%5Cfrac%7B8x%5E3-18x%7D%7B3x%2B2%7D)
The closest answer is A, so I'll go with that.
Step-by-step explanation:
So we have the rational expression:
![\frac{4x^3-9x}{2x-7}\div\frac{3x^2+2x^2}{4x^2-14x}](https://tex.z-dn.net/?f=%5Cfrac%7B4x%5E3-9x%7D%7B2x-7%7D%5Cdiv%5Cfrac%7B3x%5E2%2B2x%5E2%7D%7B4x%5E2-14x%7D)
First, remove the division sign. To do so, turn the division into multiplication and flip the second term:
![\frac{4x^3-9x}{2x-7}\cdot \frac{4x^2-14x}{3x^2+2x}](https://tex.z-dn.net/?f=%5Cfrac%7B4x%5E3-9x%7D%7B2x-7%7D%5Ccdot%20%5Cfrac%7B4x%5E2-14x%7D%7B3x%5E2%2B2x%7D)
Now, simplify. From the first term, in the numerator, factor out a x. On the second term, factor out a 2x in the numerator and a x in the denominator:
![\frac{x(4x^2-9)}{2x-7}\cdot\frac{2x(2x-7)}{x(3x+2)}](https://tex.z-dn.net/?f=%5Cfrac%7Bx%284x%5E2-9%29%7D%7B2x-7%7D%5Ccdot%5Cfrac%7B2x%282x-7%29%7D%7Bx%283x%2B2%29%7D)
Multiply straight across:
![\frac{x(4x^2-9)(2x)(2x-7)}{(2x-7)(x)(3x+2)}](https://tex.z-dn.net/?f=%5Cfrac%7Bx%284x%5E2-9%29%282x%29%282x-7%29%7D%7B%282x-7%29%28x%29%283x%2B2%29%7D)
Cancel out the (2x-7):
![\frac{x(4x^2-9)(2x)}{x(3x+2)}](https://tex.z-dn.net/?f=%5Cfrac%7Bx%284x%5E2-9%29%282x%29%7D%7Bx%283x%2B2%29%7D)
Cancel out the x:
![\frac{(2x)(4x^2-9)}{(3x+2)}](https://tex.z-dn.net/?f=%5Cfrac%7B%282x%29%284x%5E2-9%29%7D%7B%283x%2B2%29%7D)
At this point, we can factor the (4x^2-9) term, but we won't be able to cancel it out. Thus, this is the simplest it can get.
To get the answer, expand the numerator:
![=\frac{8x^3-18x}{3x+2}](https://tex.z-dn.net/?f=%3D%5Cfrac%7B8x%5E3-18x%7D%7B3x%2B2%7D)
Thus, the answer is...
A?