jmc

Let $f(x)=x^3+3x+5.$ Note that $f(x)$ is an increasing function. Furthermore, as $x$ approaches $-\infty ,$ $f(x)$ approaches $-\infty ,$ and as $x$ approaches $\infty ,$ $f(x)$ approaches $\infty .$ Therefore, the graph of $f(x)$ must cross the $x$-axis at some point (and since $f(x)$ is increasing, this point is unique), so $f(x)$ has exactly $\boxed{1}$ real root.