f(x)=-x^{5}-4x^{4}\nchoose the correct graph below.

Explanation:

Step1: Analyze the end - behavior

For the polynomial function (f(x)=-x^{5}-4x^{4}), the leading term is (-x^{5}). Since the degree (n = 5) (odd) and the leading coefficient (a=-1\lt0), as (x\to+\infty), (y = f(x)\to-\infty) and as (x\to-\infty), (y = f(x)\to+\infty).

Step2: Find the (y) - intercept

Set (x = 0) in (f(x)), then (f(0)=-(0)^{5}-4(0)^{4}=0). So the (y) - intercept is (0).

Step3: Factor the function

(f(x)=-x^{4}(x + 4)). The roots of the function are (x = 0) (with multiplicity (4)) and (x=-4) (with multiplicity (1)). At (x = 0), the graph touches the (x) - axis (because of even multiplicity) and at (x=-4) the graph crosses the (x) - axis (because of odd multiplicity).

Answer:

Based on the end - behavior, (y) - intercept, and roots analysis, we can determine the correct graph. Without seeing the actual graphs' details, we know the general shape. But if we assume the graphs are drawn with correct end - behavior and root characteristics, the graph that has (y) - intercept at (0), crosses the (x) - axis at (x=-4), touches the (x) - axis at (x = 0), goes to (+\infty) as (x\to-\infty) and to (-\infty) as (x\to+\infty) is the correct one. Since we don't have the actual visual content of the graphs A, B, C, D to choose from precisely, the above analysis gives the criteria for choosing the correct graph.