Check out the attached image below. Use polynomial long division (see figure 1) or synthetic division (figure 2) to find that the quotient is x-8
This tells us the slant asymptote is y = x-8
The presence of the slant asymptote means we do not have a horizontal asymptote. As x gets larger and larger, and heads to infinity, then y approaches the line y = x-8; rather than y approaching some single fixed value.
The vertical asymptote is x = -4 because this x value makes the denominator x+4 equal to 0. We cannot divide by zero.
Figure 3 shows the graph, which I have done using GeoGebra (a free graphing tool). This is also attached below.