Needing help finding restrictions and then simplifying

Question

6x2+11x-72

________

3x²-5x-8

Answer 1

Since the leading exponents are equal, the horizontal asymtote is at y = 6/3, or y = 2.

To find the vertical asymtote, we set the denominator equal to 0 and solve.

3x² -5x - 8 = 0

(3x - 8)(x + 1) = 0

3x - 8 = 0

3x = 8

x = 8/3

x + 1 = 0

x = -1

Now, to simplify. The numerator factors to (3x - 8)(2x + 9). 3x - 8 cancels out with the same in the denominator, so we are left with...

(2x + 9)/(x + 1)

Because 3x - 8 cancels out, we are left with a vertical asymptote at x = -1 and a hole at x = 8/3.