Multiplying the fractions results in:

f(x)=(x^2-2x-48)(9x^3-4x)/((3x^3-2x^2)(8x-64))

f(x)=x(x-8)(x+6)(9x^2-4)/(8x^2(3x-2)(x-8))

9x^2-4 = (3x)^2-2^2 = (3x+2)(3x–2)

f(x)=x(x-8)(x+6)(3x+2)(3x–2)/(8x^2(3x-2)(x-8))

f(x)=x(x-8)(3x–2)/(x(x-8)(3x–2))

* (x+6)(3x+2)/(8x)

x(x-8)(3x–2)/(x(x-8)(3x–2))

= {1 if x ≠ 0, 2/3, or 8; UNDEFINED otherwise }

I.e., there are “Holes” at x = 0, 2/3, or 8.

f(x)=(x+6)(3x+2)/(8x), x ≠ 0, 2/3, or 8

Since x is in the denominator of this reduced form there will be a vertical asymptote at x = 0.

f(x)=(3x^2+20x+12)/(8x), x ≠ 0, 2/3, or 8

Do long division of numerator by denominator:

3x^2+20x+12 | 3x/8

3x^2

––––––––––––

20x+12 | 20/8

20x

––––––

12

f(x)=3x/8 + 20/8 + 12/(8x), x ≠ 0, 2/3, or 8

f(x)=3x/8 + 5/2 + 3/(2x), x ≠ 0, 2/3, or 8

This is the simplest form; it shows a slant asymptote of y=3x/8 + 5/2.