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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.

It is given that the coefficient of x^3 in the expansion of (2 + ax)^4(5 - ax) is 432. Find the value of the constant a.

This problem explores how to find the constant a in the binomial expansion of (2 + ax)^4(5 - ax), with a focus on the coefficient of x^3 being 432. The problem uses the binomial theorem and polynomial multiplication techniques, suitable for students in Grades 10-12.

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