Answer
$\{-\frac{1}{2} \}$, $f(x)=(2x+1)(x^2+1)$
Work Step by Step
Step 1. Given $f(x)=2x^3+x^2+2x+1$, list possible rational zeros as $\frac{p}{q}=\pm1,\pm\frac{1}{2}$
Step 2. Use synthetic division as shown in the figure to find one zero $x=-\frac{1}{2}$.
Step 3. Use the quotient to solve $2x^2+2=0$ or $x^2=-1$, thus $x=\pm i$
Step 4. Thus the real zeros are $\{-\frac{1}{2} \}$ and we can factor the function as $f(x)=(2x+1)(x^2+1)$
