Answer
$\{\frac{1}{2}\}$
Work Step by Step
Step 1. Given $f(x)=x^3+\frac{3}{2}x^2+3x-2$, list possible rational zeros as $\frac{p}{q}=\pm1,\pm2$
Step 2. Use synthetic division as shown in the figure to find zero(s) $x=\frac{1}{2}$.
Step 3. Use the quotient and solve $x^2+2x+4=0$ and there are no real zeros.
Step 4. Thus the real zeros are $\{\frac{1}{2}\}$