Prove that if $x^3 + 3x + 3$ is irrational then $x$ is irrational, by proving the contrapositive I don't understand how to do this considering that the contrapositive of $x^3$ is irrational. For example $2$ to the cube root is irrational, but I am trying to prove that is is rational

A more pedestrian proof (following @Mike's first step):
Suppose $x$ is rational, and the quotient of two integers $x = \frac{a}{b}.$ Then 
\begin{align}
f(x) 
&= x^3 + 3x + 3 \\[8pt]
&= \left(\frac{a}{b}\right)^3 + 3\frac{a}{b} + 3\frac{1}{1} \\[8pt]
&= \frac{a^3}{b^3} + 3\frac{a}{b}\frac{b^2}{b^2} + 3\frac{1}{1}\frac{b^3}{b^3} \\[8pt]
&= \frac{a^3}{b^3} + 3\frac{ab^2}{b^3} + \frac{3b^3}{b^3} \\[8pt]
&= \frac{a^3 + 3ab^2 + 3b^3}{b^3}
\end{align}
which is a quotient of two integers (namely, $a^3 + 3ab^2 + 3b^2$ and $b^3$). Hence $f(x)$ is a rational as well. 
