What you want to do first is find the equation of the line. We'll put it into the form y=mx+b, where m is the slope and b is the y-intercept.
$$\frac{4}{3x+2y} = -14$$
Well, we want y all by itself, so what should we do? Let's multiply both sides of the equation by the denominator!
$$\frac{4}{3x+2y} * (3x+2y) = -14 * (3x+2y) \Rightarrow 4 = -42x - 28y$$
I went ahead and used the distributive property to help simplify the problem. Now let's keep making y alone! So, y doesn't like x, so we'll take him away by adding 42x to each side of the equation.
$$4 + 42x = -42x - 28x +42x \Rightarrow 4 + 42x = -28y$$
The y still isn't alone. He has that -28 right next to him! Let's divide both sides by -28 to get rid of him.
$$\frac{4+42x}{-28} = \frac{-28y}{-28} \Rightarrow -\frac{1}{7} - \frac{3}{2}x = y$$
Now y is alone! let's rewrite this as the whole y=mx+b form.
$$y = -\frac{3}{2}x-\frac{1}{7}$$
Remember, m is the slope. So, m=-3/2!
Now, to do the perpendicular of a line, just flip the slope and change the sign! So, the new perpendicular slope equals 2/3!