Let's work through this step by step.

Part a) What is the value of $f(1)$?

We are given the function $f(x) = x^3 + 7x^2 + 4x - 12$.

To find $f(1)$, substitute $x = 1$ into the function:

$f(1) = (1)^3 + 7(1)^2 + 4(1) - 12$ $f(1) = 1 + 7 + 4 - 12$ $f(1) = 0$

So, $f(1) = 0$.

Part b) Using your answer to part a), select a factor of $f(x)$.

Since $f(1) = 0$, this means that $(x - 1)$ is a factor of $f(x)$ (by the factor theorem, if $f(a) = 0$, then $(x - a)$ is a factor of $f(x)$).

Thus, the correct factor from the options is $(x - 1)$.

Do you want more details or have any questions on this?

Here are 5 follow-up questions to deepen your understanding:

1. How can the Factor Theorem help us identify other factors of $f(x)$?
2. Can we factor $f(x)$ completely, given that $(x - 1)$ is a factor?
3. What are other possible methods to factor cubic polynomials like $f(x)$?
4. How would you check if $(x + 1)$ or $(x - 12)$ is a factor of $f(x)$?
5. How would you solve $f(x) = 0$ for the remaining roots?

Tip: When dealing with polynomials, always check if substituting values into the function gives 0 to find possible factors using the Factor Theorem.