Answer
$x=-1,-\frac{1}{3}$.
$f(x) =(3x+1)(x+1)(x^2+2)$
Work Step by Step
Step 1. For $f(x)=3x^4+4x^3+7x^2+8x+2$, list possible rational real zeros $\frac{p}{q}: \pm1,\pm2,\pm\frac{1}{3},\pm\frac{2}{3}$
Step 2. Use synthetic division as shown in the figure to find a zero(s) $x=-1,-\frac{1}{3}$.
Step 3. Use the quotient to find other zeros: $3x^2+6=0$ no real solutions.
Step 4. Factor the polynomial $f(x)=(x+\frac{1}{3})(x+1)(3x^2+6)=(3x+1)(x+1)(x^2+2)$
