The given function is:

$f(x) = 2x^5 + x^4 - 18x^3 - 17x^2 + 20x + 12$

It is stated that $x - 3$ is a factor of $f(x)$. By the Factor Theorem, if $x - 3$ is a factor, then $f(3) = 0$. Let's verify that by calculating $f(3)$.

I'll compute $f(3)$ by substituting $x = 3$ into the function:

$f(3) = 2(3)^5 + (3)^4 - 18(3)^3 - 17(3)^2 + 20(3) + 12$

Let me calculate this value.The value of $f(3)$ is 0, confirming that $x - 3$ is indeed a factor of $f(x)$, as expected from the Factor Theorem.

Would you like further details or have any other questions?

Here are five related questions you might find useful:

1. How does the Factor Theorem help in finding factors of a polynomial?
2. Can $f(x)$ have other factors besides $x - 3$?
3. How would you completely factor $f(x)$?
4. What happens if we substitute other values of $x$ into $f(x)$?
5. How can we use synthetic division to confirm $x - 3$ is a factor?

Tip: When testing potential factors of a polynomial, always substitute the corresponding root into the function to verify if it results in zero.