This type of problem usually requires putting in some values for the x and seeing what pops out. However, if that takes too long, using some precalculus will help. What do I mean by that?
Take for example the equation we are given (x-3)/(x+4). The horizontal tangent is the limit of your equation as x goes to infinity or negative infinity. In this, case it is just 1.
I can tell because both numerator and denominator have a degree of 1(i.e. x1) and so as I approach infinity they will grow by the same amount, namely linearly. Furthermore, each linear expression has 1 as a coefficient, which indicates the rate of growth is 1 for both examples. Thus 1/1=1.
Now, the vertical tangent occurs when the denominator is equal to 0. So x+4=0 implies x=-4 is a vertical asymptote.
Finally, when you plug in the x values to find the slopes, you will notice that they give one value for the slope for each x value. There is no y value in your slope. Thus, the slope is independent of the position of the y value. Therefore, for each value of x, there will be a single slope for all the values of y. So, the slope field will be parallel along each column since each x value will generate one series of slopes.
So, the answer choice as you indicated would be 4) II and III.