Answer
$\{-\frac{1}{2}, 1, \pm \sqrt 2 \}$, $f(x)=(x-1)(2x+1)(x+\sqrt 2)(x-\sqrt 2)$
Work Step by Step
Step 1. Given $f(x)=2x^4-x^3-5x^2+2x+2$, list possible rational zeros as $\frac{p}{q}=\pm1,\pm2,\pm\frac{1}{2}$
Step 2. Use synthetic division as shown in the figure to find two zero $x=1,-\frac{1}{2}$.
Step 3. Use the quotient to solve $2x^2-4=0$ or $x^2=2$, thus $x=\pm \sqrt 2$
Step 4. Thus the real zeros are $\{-\frac{1}{2}, 1, \pm \sqrt 2 \}$ and we can factor the function as $f(x)=(x-1)(2x+1)(x+\sqrt 2)(x-\sqrt 2)$