Answer
$-3,2, \pm i\frac{\sqrt 2}{2}$
$f(x)=2(x+ i\frac{\sqrt 2}{2})(x- i\frac{\sqrt 2}{2})(x+3)(x-2)$
Work Step by Step
Step 1. Based on the Rational Zeros Theorem, list possible rational zeros $\frac{p}{q}=\pm1,\pm2,\pm3,\pm6,\pm\frac{1}{2},\pm\frac{3}{2}$
Step 2. Use synthetic division to find one or more zeros $x=2,-3$ as shown in the figure.
Step 3. Use to the quotient to solve $2x^2+1=0$ or $x^2=-\frac{1}{2}$, thus $x=\pm i\frac{\sqrt 2}{2}$
Step 4. We can factor the function as $f(x)=2(x+ i\frac{\sqrt 2}{2})(x- i\frac{\sqrt 2}{2})(x+3)(x-2)$
