Ok, let's solve this equation step by step.

First, let's get rid of the fraction by cross-multiplying:

(3x + 8) = (x - 4)(2x + 1)

Now, let's expand the right side of the equation:

3x + 8 = 2x^2 + x - 8

Rearrange the terms to set the equation equal to zero:

2x^2 - 2x - 16 = 0

Now, let's solve for x by factoring:

2x^2 - 2x - 16 = 0
2(x^2 - x - 8) = 0
2(x - 4)(x + 2) = 0

This gives us two possible solutions:

x = 4
x = -2

Therefore, the solutions to the equation (3x+8)/(2x+1) = x-4 are x = 4 and x = -2.