Chapter 3 - Polynomial and Rational Functions - Section 3.2 The Real Zeros of a Polynomial Function - 3.2 Assess Your Understanding - Page 224: 58

$x=-\frac{3}{2}$.

Step 1. For $2x^3+3x^2+2x+3=0$, list possible rational real zeros $\frac{p}{q}: \pm1,\pm3,\pm\frac{1}{2},\pm\frac{3}{2}$ Step 2. Use synthetic division as shown in the figure to find zero(s) $x=-\frac{3}{2}$. Step 3. Use the quotient to find other zeros: $2x^2+2=0$ no real solutions.