The given function is:

$f(x) = 2x^3 + 2x$

Would you like to find its derivative, integral, or evaluate it for specific values? Let me know how you'd like to proceed!

Here are five related questions you might want to explore:

1. What is the first derivative of $f(x) = 2x^3 + 2x$?
2. How would you find the critical points of this function?
3. What is the integral of $f(x)$?
4. How does the function behave as $x \to \infty$?
5. What are the roots of the equation $f(x) = 0$?

Tip: For polynomial functions, the degree of the polynomial indicates how many real roots (including complex and repeated roots) it may have.