We are tasked with finding the constant $a$ in the expansion of $(2 + ax)^4(5 - ax)$ given that the coefficient of $x^3$ is 432.

Step-by-Step Solution:

1. Expression Breakdown: We need to expand the expression $(2 + ax)^4(5 - ax)$, focusing only on the terms that contribute to $x^3$.

2. Binomial Expansion: Use the binomial expansion for both parts of the expression:

   • $(2 + ax)^4 = \sum_{k=0}^4 \binom{4}{k} 2^{4-k} (ax)^k$
   • $(5 - ax) = 5 - ax$

3. Finding Terms with $x^3$: Multiply the relevant terms from $(2 + ax)^4$ with $(5 - ax)$, ensuring the total power of $x$ is 3.

   • From $(2 + ax)^4$, the term with $x^3$ is: $\binom{4}{3} 2^{4-3} (ax)^3 = 4 \cdot 2 \cdot a^3x^3 = 8a^3x^3$
   • From $(5 - ax)$, pick the constant term $5$, so the contribution is $8a^3x^3 \cdot 5 = 40a^3x^3$.

   Another contribution to $x^3$ comes from the term $\binom{4}{2} 2^{4-2} (ax)^2$ in $(2 + ax)^4$, which is: $\binom{4}{2} 2^{4-2} (ax)^2 = 6 \cdot 4 \cdot a^2x^2 = 24a^2x^2$ Now, multiply this by the $-ax$ term from $(5 - ax)$: $24a^2x^2 \cdot (-ax) = -24a^3x^3$

4. Summing the Contributions to $x^3$: The total contribution to $x^3$ is: $40a^3x^3 - 24a^3x^3 = 16a^3x^3$

5. Equating to Given Coefficient: We are told that the coefficient of $x^3$ is 432, so: $16a^3 = 432$ Solving for $a^3$: $a^3 = \frac{432}{16} = 27$ Therefore: $a = \sqrt[3]{27} = 3$

Thus, the value of $a$ is $\boxed{3}$.

Do you have any questions or need more details on any part of this solution? Here are some related questions you can explore:

1. What is the general formula for binomial expansion?
2. How do we identify the required terms in a binomial expansion?
3. What role does the power of $x$ play in identifying the coefficient?
4. Can we use alternative methods to solve for $a$, like factoring?
5. How do higher powers of $a$ affect the expansion result?

Tip: Always ensure you focus on terms that match the power of the variable you're interested in when using binomial expansion.