According to the factor theorem, if (x-c) is a factor of f(x), then f(c)=0. Here, c=4.

Here are the solutions to the questions:

a) Use the factor theorem to show that $(x-4)$ is a factor of $f(x) = 2x^3 - x^2 - 25x - 12$.

Step 1: According to the factor theorem, if $(x-c)$ is a factor of $f(x)$, then $f(c)=0$. Here, $c=4$.

Step 2: Substitute $x=4$ into $f(x)$:

$$f(4) = 2(4)^3 - (4)^2 - 25(4) - 12$$ $$$$