Answer
$x=-2,-1$, $x=2$ (multiplicity 2).
$f(x)=(x+2)(x+1)(x-2)^2$
Work Step by Step
Step 1. For $f(x)=x^4-x^3-6x^2+4x+8$, list possible rational real zeros $\frac{p}{q}: \pm1,\pm2,\pm4,\pm8$
Step 2. Use synthetic division as shown in the figure to find a zero(s) $x=-2,-1$.
Step 3. Use the quotient to find other zeros: $x^2-4x+4=0\Longrightarrow (x-2)^2=0 \Longrightarrow x=2$ (multiplicity 2).
Step 4. Factor the polynomial $f(x)=(x+2)(x+1)(x-2)^2$
