The problem provided involves finding the coefficient of a term in the binomial expansion of $2x^2(3 + kx)^7$ to solve for $k$. Here’s a breakdown of the steps:

1. Identify the Term in Expansion:

   • In the expansion of $(3 + kx)^7$, we are asked to find the term where $x$ has an exponent of 5.
   • For binomial expansions, the general term is given by: $T_{r+1} = \binom{7}{r} \cdot 3^{7-r} \cdot (kx)^r$
   • Since we need the term where $x$ has a power of 5, we set $r = 5$.

2. Substitute $r = 5$:

   • The term involving $x^5$ becomes: $T_6 = \binom{7}{5} \cdot 3^{7-5} \cdot (kx)^5$
   • Calculating each part:
     • $\binom{7}{5} = 21$
     • $3^2 = 9$
     • $(kx)^5 = k^5 x^5$
   • Therefore, the coefficient of $x^5$ is: $21 \cdot 9 \cdot k^5 = 189k^5$

3. Set the Coefficient Equal to 210:

   • According to the problem, this coefficient is given as 210: $189k^5 = 210$

4. Solve for $k$:

   • Dividing both sides by 189: $k^5 = \frac{210}{189} = \frac{70}{63} = \frac{10}{9}$
   • Taking the fifth root: $k = \sqrt[5]{\frac{10}{9}}$

The provided solution then simplifies $\sqrt[5]{\frac{10}{9}}$ to arrive at a final answer for $k$.

Would you like further explanation on any specific step?

Here are five related questions to deepen understanding:

1. How would the solution change if the power in the binomial expansion were different?
2. Why is the fifth root used in the solution process?
3. How can we verify if the solution for $k$ is correct?
4. What does the binomial theorem state for general expansions?
5. How would a coefficient change if different terms in $x$ were requested?

Tip: Always check that the powers and terms align with the term you're finding in a binomial expansion for accuracy.