The given polynomial function is:

$P(x) = x^4 - 3x^3 - 3x^2 + 7x + 6$

To solve or analyze this, it seems like you want to graph the function or find key characteristics such as roots, critical points, and behavior of the graph. Here are the steps we can take:

1. Find the roots of $P(x)$: Determine the values of $x$ where $P(x) = 0$. This involves factoring or using numerical methods if necessary.
2. Identify the critical points: These occur where the derivative, $P'(x)$, is zero or undefined.
3. Analyze the concavity: Use the second derivative, $P''(x)$, to find points of inflection and determine intervals of concavity.
4. Evaluate end behavior: Since $P(x)$ is a fourth-degree polynomial with a positive leading coefficient, the end behavior will go to positive infinity as $x \to \pm\infty$.
5. Graph the function: Plot the polynomial using the information gathered.

Would you like me to proceed with finding the roots and derivatives, or is there a specific aspect of the problem you're interested in?

Here are five related questions:

1. How do you find the derivative of a polynomial function?
2. What methods can be used to factor higher-degree polynomials?
3. How do the roots of a polynomial relate to its graph?
4. What is the significance of critical points in a function?
5. How does the degree of a polynomial affect its graph's shape?

Tip: When graphing a polynomial, knowing the roots, critical points, and end behavior can give you a good idea of its general shape without needing to plot numerous points.