To determine the degree of the polynomial $f(x) = x^2 (x - 3)^a (x + 2)^3 (x + 1)^b$ with $a = 4$ and $b = 5$, we can follow these steps:

1. Substitute the values of $a$ and $b$: $f(x) = x^2 (x - 3)^4 (x + 2)^3 (x + 1)^5$

2. Identify the degree of each term:

   • The term $x^2$ has a degree of $2$.
   • The term $(x - 3)^4$ has a degree of $4$.
   • The term $(x + 2)^3$ has a degree of $3$.
   • The term $(x + 1)^5$ has a degree of $5$.

3. Add the degrees: The degree of the polynomial $f(x)$ is the sum of the degrees of all the terms: $2 + 4 + 3 + 5 = 14$

Thus, the degree of the polynomial $f(x)$ is 14.

Would you like additional details or have any questions?

Here are 5 related questions to deepen understanding:

1. How does the degree change if the exponents $a$ and $b$ are different?
2. What is the leading coefficient of the polynomial if expanded fully?
3. How does the degree of a polynomial affect its behavior as $x \to \infty$ or $x \to -\infty$?
4. Can the degree of a polynomial ever be a non-integer?
5. How does the degree relate to the number of roots a polynomial might have?

Tip: The degree of a polynomial is the sum of the exponents of its highest-power terms in each factor.