Chapter 4 - Polynomial and Rational Functions - 4.5 The Real Zeros of a Polynomial Function - 4.5 Assess Your Understanding - Page 233: 58

$\{-\frac{3}{2}\}$

Step 1. Given $f(x)=2x^3+3x^2+2x+3$, list possible rational zeros as $\frac{p}{q}=\pm1,\pm3,\pm\frac{1}{2},\pm\frac{3}{2}$ Step 2. Use synthetic division as shown in the figure to find a zero $x=-\frac{3}{2}$. Step 3. Use the quotient to solve $2x^2+2=0$ or $x^2=-1$, thus $x=\pm i$ Step 4. Thus the real zeros are $\{-\frac{3}{2}\}$