The problem asks to find the value of $k$ given that $3x^3 + kx + 10$ is divisible by $x + 2$.

We are told that $x + 2$ is a factor of $f(x) = 3x^3 + kx + 10$. This means that when we substitute $x = -2$ into the polynomial $f(x)$, the result must be zero (according to the factor theorem).

The function $f(x) = 3x^3 + kx + 10$.

Substituting $x = -2$:

$f(-2) = 3(-2)^3 + k(-2) + 10 = 0$

Simplify each term:

$3(-8) + (-2k) + 10 = 0$

$-24 - 2k + 10 = 0$

$-14 - 2k = 0$

Solve for $k$:

$-2k = 14 \quad \Rightarrow \quad k = -7$

Thus, the value of $k$ is $\boxed{-7}$.

Do you want more details on this solution? Here are some follow-up questions:

1. What is the Factor Theorem and how is it used to solve problems like this one?
2. How does synthetic division relate to finding factors of polynomials?
3. Can $x + 2$ have any other roots in this cubic equation?
4. How would the equation change if $x + 2$ was not a factor?
5. Can this type of method be used for polynomials of degree higher than three?

Tip: The Factor Theorem is very powerful for checking if a linear factor divides a polynomial without fully dividing the polynomial!