The function provided in the image is:

$f(x) = x^4 + 4x^3 + 6$

Would you like me to find:

1. The first derivative $f'(x)$ for critical points or slope analysis?
2. The second derivative $f''(x)$ for concavity or inflection points?
3. The roots or values of $f(x)$ at specific points?
4. The behavior of $f(x)$ as $x$ approaches large positive or negative values?
5. The local minima, maxima, or other characteristics of this polynomial function?

Tip: To analyze a polynomial function, understanding its derivatives helps in determining the behavior and shape of its graph.