Answer
$\{-1,-\frac{1}{3} \}$, $f(x)=(3x+1)(x+1)(x^2+2)$
Work Step by Step
Step 1. Given $f(x)=3x^4+4x^3+7x^2+8x+2$, list possible rational zeros as $\frac{p}{q}=\pm1,\pm2,\pm\frac{1}{3},\pm\frac{2}{3}$
Step 2. Use synthetic division as shown in the figure to find two zero $x=-1,-\frac{1}{3}$.
Step 3. Use the quotient to solve $3x^2+6=0$ or $x^2=-2$, thus $x=\pm\sqrt 2 i$
Step 4. Thus the real zeros are $\{-1,-\frac{1}{3} \}$ and we can factor the function as $f(x)=(3x+1)(x+1)(x^2+2)$
