Answer
$f(x)=\frac{1}{5}(x+4)(x+1)^2(x-3)$
Work Step by Step
Step 1. Based on the given graph, we can identify the real zeros as $x=-4$ (multiplicity 1), $x=-1$ (multiplicity 2), and $x=3$ (multiplicity 1).
Step 2. We can write a general form of the polynomial as $f(x)=a(x+4)(x+1)^2(x-3)$ where $a$ is unknown.
Step 3. Use the given point $(1,-8)$ on the curve, we have $f(1)=a(1+4)(1+1)^2(1-3)=-8$, thus $a=\frac{1}{5}$
Step 4. Thus the function is $f(x)=\frac{1}{5}(x+4)(x+1)^2(x-3)$
