(6x^3-8x^2-6x)/(3-11x-8x^3)

nothing to see here

well, there is a horizontal asymptote at y = -3/4

To simplify the expression (6x^3 - 8x^2 - 6x) / (3 - 11x - 8x^3), we can follow these steps:

Step 1: Factor both the numerator and denominator:
In the numerator:
6x^3 - 8x^2 - 6x = 2x(3x^2 - 4x - 3)
In the denominator:
3 - 11x - 8x^3 = -(8x^3 + 11x - 3)

Step 2: Re-write the denominator with the terms in descending order of their exponents:
-(8x^3 + 11x - 3) = -8x^3 - 11x + 3

Step 3: Divide the numerator by the denominator. To do this, we can use polynomial long division or synthetic division. Let's use polynomial long division:

2x + 1
___________
(-8x^3 - 11x + 3) | 2x(3x^2 - 4x - 3)

Step 4: Perform the polynomial long division:

First, divide the highest-degree terms:
2x * -8x^3 = -16x^4

-16x^4
_____________
(-8x^3 - 11x + 3)

Next, multiply the divisor (2x) by each term of the quotient (-16x^4):
-16x^4 * -8x^3 = 128x^7
-16x^4 * -11x = 176x^5
-16x^4 * 3 = -48x^4

2x + 1
___________
(-8x^3 - 11x + 3) | 2x(3x^2 - 4x - 3)
+ 128x^7 + 176x^5 - 48x^4

Next, subtract the product from the dividend:

2x + 1
___________
(-8x^3 - 11x + 3) | 2x(3x^2 - 4x - 3)
+ 128x^7 + 176x^5 - 48x^4
-(-16x^4 + 22x^4 - 6x^4 + 22x^4 - 30x^4 + 8x^4 - 176x^5 + 16x^5 - 48x^4)

Simplifying the subtraction:
2x + 1
___________
(-8x^3 - 11x + 3) |2x(3x^2 - 4x - 3)
+ 128x^7 + 176x^5 - 48x^4
+ 16x^4 - 176x^5 + 16x^5 + 48x^4
_____________________
128x^7 + 40x^5

At this point, we have divided the numerator by the denominator. The simplified expression is:

(6x^3 - 8x^2 - 6x) / (3 - 11x - 8x^3) = 2x + 1 + (128x^7 + 40x^5) / (3 - 11x - 8x^3)

Please note that this might not be the final simplified expression as the division process could continue with the remaining terms.