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To solve this problem, we need to find the remainder when the polynomial \( f(x) = x^3 + 4ax^2 + 7x + 2a \) is divided by \( x - \frac{5a}{3} \).

By the **Remainder Theorem**, if a polynomial \( f(x) \) is divided by \( x - k \), the remainder is simply \( f(k) \). Here, \( k = \frac{5a}{3} \), so we need to evaluate \( f\left(\frac{5a}{3}\right) \).

### Step-by-Step Solution

1. **Substitute \( x = \frac{5a}{3} \) into \( f(x) \):**

   \[
   f\left(\frac{5a}{3}\right) = \left(\frac{5a}{3}\right)^3 + 4a\left(\frac{5a}{3}\right)^2 + 7\left(\frac{5a}{3}\right) + 2a
   \]

2. **Calculate each term individually:**

   - The first term: \(\left(\frac{5a}{3}\right)^3 = \frac{125a^3}{27}\)
   - The second term: \(4a\left(\frac{5a}{3}\right)^2 = 4a \cdot \frac{25a^2}{9} = \frac{100a^3}{9}\)
   - The third term: \(7\left(\frac{5a}{3}\right) = \frac{35a}{3}\)
   - The fourth term: \(2a\)

3. **Combine these terms:**

   \[
   f\left(\frac{5a}{3}\right) = \frac{125a^3}{27} + \frac{100a^3}{9} + \frac{35a}{3} + 2a
   \]

4. **Express all terms with a common denominator (27) and simplify:**

   - The first term is already \(\frac{125a^3}{27}\).
   - The second term: \(\frac{100a^3}{9} = \frac{300a^3}{27}\)
   - The third term: \(\frac{35a}{3} = \frac{315a}{27}\)
   - The fourth term: \(2a = \frac{54a}{27}\)

   Now we can rewrite the expression as:

   \[
   f\left(\frac{5a}{3}\right) = \frac{125a^3 + 300a^3 + 315a + 54a}{27}
   \]

5. **Combine like terms:**

   \[
   f\left(\frac{5a}{3}\right) = \frac{425a^3 + 369a}{27}
   \]

6. **Express the result in the required form** \( \frac{A}{B} + \frac{C}{D} \):

   We can split this as:

   \[
   f\left(\frac{5a}{3}\right) = \frac{425a^3}{27} + \frac{369a}{27} = \frac{425a^3}{27} + \frac{123a}{9}
   \]

Therefore, the remainder when \( x^3 + 4ax^2 + 7x + 2a \) is divided by \( x - \frac{5a}{3} \) is:

\[
\frac{425a^3}{27} + \frac{123a}{9}
\]

Would you like a detailed breakdown of each step, or do you have any specific questions?

### Further Questions
1. How is the Remainder Theorem used in polynomial division?
2. Can this method be applied to any polynomial division problem?
3. What are alternative ways to find remainders in polynomial division?
4. How would the result change if \( a = 1 \) or any specific value?
5. What happens if the divisor has a higher degree?

### Tip
Always simplify terms with a common denominator early to reduce calculation complexity.

Find the remainder when x^3 + 4ax^2 + 7x + 2a is divided by x - 5a/3. Give your answer in form A/B + C/D.

Learn how to find the remainder when the polynomial x^3 + 4ax^2 + 7x + 2a is divided by x - 5a/3 using the Remainder Theorem. This solution provides a step-by-step guide for students in Grades 10-12 on polynomial division and the application of the Remainder Theorem.

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