Find the number of real roots of
\[2x^{2001} + 3x^{2000} + 2x^{1999} + 3x^{1998} + \dots + 2x + 3 = 0.\]
We can factor the given equation as
\[(2x + 3) x^{2000} + (2x + 3) x^{1998} + \dots + (2x + 3) = (2x + 3)(x^{2000} + x^{1998} + \dots + 1) = 0.\]Thus, $x = -\frac{3}{2}$ is a root.  Note that
\[x^{2000} + x^{1998} + \dots + 1 \ge 1\]for all real $x,$ so the given polynomial has only $\boxed{1}$ real root.